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Precalculus Quiz

Precalculus Quiz: Matrix Addition Subtraction And Multiplication

Practice Matrix Addition Subtraction And Multiplication in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

Let AAA and BBB be 2×22\times 22×2 matrices: A=[1021],B=[0110].A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}.A=[12​01​],B=[01​10​]. Given the matrices above, which statement about ABABAB and BABABA is true?​​

Select an answer to continue

What this quiz covers

This quiz focuses on Matrix Addition Subtraction And Multiplication, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let AAA and BBB be 2×22\times 22×2 matrices: A=[1021],B=[0110].A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}.A=[12​01​],B=[01​10​]. Given the matrices above, which statement about ABABAB and BABABA is true?​​

  1. AB=BAAB=BAAB=BA
  2. AB≠BAAB\neq BAAB=BA (correct answer)
  3. ABABAB is not defined, but BABABA is defined.
  4. BABABA is not defined, but ABABAB is defined.

Explanation: This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB ≠ BA, even when both products are defined. Computing AB gives [[0, 1], [1, 2]] while computing BA gives [[2, 1], [1, 0]], demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because it shows AB ≠ BA with the specific matrices from the stimulus. Choice A assumes AB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB ≠ BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products.

Question 2

Let AAA be a 2×22 \times 22×2 matrix and BBB be a 2×22 \times 22×2 matrix: A=[210−3],B=[−1450].A=\begin{bmatrix}2&1\\0&-3\end{bmatrix},\quad B=\begin{bmatrix}-1&4\\5&0\end{bmatrix}.A=[20​1−3​],B=[−15​40​]. Using matrices AAA and BBB, what is the entry in row 222, column 111 of the matrix product ABABAB? (Rows and columns are 1-indexed.)

  1. −15-15−15 (correct answer)
  2. 151515
  3. −5-5−5
  4. 555

Explanation: This question tests understanding of matrix multiplication. Matrix multiplication of an m×nm \times nm×n matrix A with an n×pn \times pn×p matrix B produces an m×pm \times pm×p matrix AB, where entry (i,j) equals the dot product of row i of A with column j of B. To find entry (2,1) of AB, we take the dot product of row 2 of A (which is [0,−3][0, -3][0,−3]) with column 1 of B (which is [−1,5][-1, 5][−1,5]), computing 0∗(−1)+(−3)∗5=−150*(-1) + (-3)*5 = -150∗(−1)+(−3)∗5=−15. Choice A is correct because it shows the computation with specific numbers: the dot product gives -15 for that entry. Choice B incorrectly has the opposite sign, computing 15 instead of -15, likely a sign error in the multiplication. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 3

Let AAA and BBB be 2×22\times 22×2 matrices: A=[2−103],B=[−4510].A=\begin{bmatrix}2&-1\\0&3\end{bmatrix},\quad B=\begin{bmatrix}-4&5\\1&0\end{bmatrix}.A=[20​−13​],B=[−41​50​]. Given the matrices above, what is A+BA+BA+B?

  1. [−2413]\begin{bmatrix}-2&4\\1&3\end{bmatrix}[−21​43​] (correct answer)
  2. [−2613]\begin{bmatrix}-2&6\\1&3\end{bmatrix}[−21​63​]
  3. [6−6−13]\begin{bmatrix}6&-6\\-1&3\end{bmatrix}[6−1​−63​]
  4. [−8−513]\begin{bmatrix}-8&-5\\1&3\end{bmatrix}[−81​−53​]

Explanation: This question tests understanding of matrix addition. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A±B)ij=Aij±Bij(A \pm B)_{ij} = A_{ij} \pm B_{ij}(A±B)ij​=Aij​±Bij​. To compute A + B, we add corresponding entries: the entry in row 1, column 1 of the result is obtained by adding the entry in row 1, column 1 of A with the same-position entry of B, giving 2+(−4)=−22 + (-4) = -22+(−4)=−2. Choice A is correct because it shows the computation with specific numbers: [2+(−4)−1+50+13+0]=[−2413]\begin{bmatrix} 2 + (-4) & -1 + 5 \\ 0 + 1 & 3 + 0 \end{bmatrix} = \begin{bmatrix} -2 & 4 \\ 1 & 3 \end{bmatrix}[2+(−4)0+1​−1+53+0​]=[−21​43​]. Choice B makes an arithmetic error in entry (1,2), computing −1+5=4-1 + 5 = 4−1+5=4 but listing 6 instead of the correct 4. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 4

Let CCC and DDD be 2×32\times 32×3 matrices: C=[10−23−14],D=[−3250−41].C=\begin{bmatrix}1&0&-2\\3&-1&4\end{bmatrix},\quad D=\begin{bmatrix}-3&2&5\\0&-4&1\end{bmatrix}.C=[13​0−1​−24​],D=[−30​2−4​51​]. Based on the matrices defined, what is C−DC-DC−D?

  1. [−2233−55]\begin{bmatrix}-2&2&3\\3&-5&5\end{bmatrix}[−23​2−5​35​]
  2. [4−2−7333]\begin{bmatrix}4&-2&-7\\3&3&3\end{bmatrix}[43​−23​−73​] (correct answer)
  3. [−2−2−73−53]\begin{bmatrix}-2&-2&-7\\3&-5&3\end{bmatrix}[−23​−2−5​−73​]
  4. [2−2−3−35−5]\begin{bmatrix}2&-2&-3\\-3&5&-5\end{bmatrix}[2−3​−25​−3−5​]

Explanation: This question tests understanding of matrix subtraction. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ. To compute C - D, we subtract corresponding entries: the entry in row 1, column 1 of the result is obtained by subtracting the entry in row 1, column 1 of D from the same-position entry of C, giving 1 - (-3) = 4. Similarly, we get 0 - 2 = -2 for position (1,2), -2 - 5 = -7 for position (1,3), 3 - 0 = 3 for position (2,1), -1 - (-4) = 3 for position (2,2), and 4 - 1 = 3 for position (2,3). Choice B is correct because it shows the result matrix [[4, -2, -7], [3, 3, 3]] obtained by subtracting corresponding entries. Choice C has a sign error in the entry (1,1), computing 1 + (-3) = -2 instead of 1 - (-3) = 4, likely from incorrectly handling the double negative. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 5

Let AAA and BBB be 2×22\times 22×2 matrices: A=[2−103],B=[−451−2].A=\begin{bmatrix}2&-1\\0&3\end{bmatrix},\quad B=\begin{bmatrix}-4&5\\1&-2\end{bmatrix}.A=[20​−13​],B=[−41​5−2​]. Given the matrices above, what is A+BA+BA+B?

  1. [−2411]\begin{bmatrix}-2&4\\1&1\end{bmatrix}[−21​41​] (correct answer)
  2. [−2611]\begin{bmatrix}-2&6\\1&1\end{bmatrix}[−21​61​]
  3. [−24−11]\begin{bmatrix}-2&4\\-1&1\end{bmatrix}[−2−1​41​]
  4. [8−5−15]\begin{bmatrix}8&-5\\-1&5\end{bmatrix}[8−1​−55​]

Explanation: This question tests understanding of matrix addition. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ. To compute A + B, we add corresponding entries: the entry in row 1, column 1 of the result is obtained by adding the entry in row 1, column 1 of A with the same-position entry of B, giving 2 + (-4) = -2. Similarly, we get -1 + 5 = 4 for position (1,2), 0 + 1 = 1 for position (2,1), and 3 + (-2) = 1 for position (2,2). Choice A is correct because it shows the result matrix [[-2, 4], [1, 1]] obtained by adding corresponding entries. Choice D incorrectly subtracts B from A instead of adding, computing A - B rather than A + B. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match.

Question 6

Matrices PPP and QQQ are given below: P=[12−10],Q=[3−241].P=\begin{bmatrix}1&2\\-1&0\end{bmatrix},\quad Q=\begin{bmatrix}3&-2\\4&1\end{bmatrix}.P=[1−1​20​],Q=[34​−21​]. Using matrices PPP and QQQ, what is the product PQPQPQ?

  1. [110−32]\begin{bmatrix}11&0\\-3&2\end{bmatrix}[11−3​02​] (correct answer)
  2. [1103−2]\begin{bmatrix}11&0\\3&-2\end{bmatrix}[113​0−2​]
  3. [3−4−40]\begin{bmatrix}3&-4\\-4&0\end{bmatrix}[3−4​−40​]
  4. [11−4−3−2]\begin{bmatrix}11&-4\\-3&-2\end{bmatrix}[11−3​−4−2​]

Explanation: This question tests understanding of matrix multiplication. Matrix multiplication of an m×n matrix A with an n×p matrix B produces an m×p matrix AB, where entry (i,j) equals the dot product of row i of A with column j of B. To find entry (1,1) of PQ, we take the dot product of row 1 of P (which is [1,2]) with column 1 of Q (which is [3,4]), computing 13 + 24 = 3 + 8 = 11. Choice A is correct because it shows the computation with specific numbers: [[13 + 24, 1*(-2) + 21],[-13 + 04, -1(-2) + 01]] = [[11,0],[-3,2]]. Choice D makes an arithmetic error in entry (1,2), computing 1(-2) + 2*1 = -2 + 2 = 0 but listing -4 instead of the correct 0. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products. Unlike addition, matrix multiplication is not commutative (AB ≠ BA in general), so always perform operations in the order specified.

Question 7

Let AAA and BBB be 2×22\times 22×2 matrices given by A=[120−1],B=[3045].A=\begin{bmatrix}1&2\\0&-1\end{bmatrix},\quad B=\begin{bmatrix}3&0\\4&5\end{bmatrix}.A=[10​2−1​],B=[34​05​]. Based on the matrices defined above, which statement about ABABAB and BABABA is true?​​

  1. AB=BAAB=BAAB=BA
  2. AB≠BAAB\neq BAAB=BA (correct answer)
  3. ABABAB is undefined but BABABA is defined
  4. BABABA is undefined but ABABAB is defined

Explanation: This question tests understanding of the non-commutative nature of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB ≠ BA, even when both products are defined. Computing AB gives [[1,2],[0,-1]]·[[3,0],[4,5]] = [[11,10],[-4,-5]], while computing BA gives [[3,0],[4,5]]·[[1,2],[0,-1]] = [[3,6],[4,3]], demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because AB and BA produce different results: AB = [[11,10],[-4,-5]] while BA = [[3,6],[4,3]]. Choice A incorrectly assumes AB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB ≠ BA in general), so always perform operations in the order specified.

Question 8

Let AAA be a 2×32 \times 32×3 matrix and BBB be a 3×23 \times 23×2 matrix: A=[20−1314],B=[1−205−31].A=\begin{bmatrix}2&0&-1\\3&1&4\end{bmatrix},\quad B=\begin{bmatrix}1&-2\\0&5\\-3&1\end{bmatrix}.A=[23​01​−14​],B=​10−3​−251​​. Given the matrices above, what are the dimensions of the matrix ABABAB?

  1. 2×22 \times 22×2 (correct answer)
  2. 3×33 \times 33×3
  3. 2×32 \times 32×3
  4. 3×23 \times 23×2

Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the number of columns in A to match the number of rows in B, and since A is 2×32 \times 32×3 and B is 3×23 \times 23×2, the operation is defined. Choice A is correct because (2×3)(3×2)→2×2(2 \times 3)(3 \times 2) \to 2 \times 2(2×3)(3×2)→2×2 gives the dimensions of AB as 2×22 \times 22×2. Choice C incorrectly states the result dimensions as 2×32 \times 32×3 when (2×3)(3×2)→2×2(2 \times 3)(3 \times 2) \to 2 \times 2(2×3)(3×2)→2×2 gives the correct dimensions. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×n and B is n×p, result is m×p. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match.

Question 9

A manufacturer produces two products using three raw materials. The usage matrix UUU shows units of each raw material needed per product, and the cost matrix CCC shows the cost per unit of each raw material. Given U=[253142]U = \begin{bmatrix} 2 & 5 \\ 3 & 1 \\ 4 & 2 \end{bmatrix}U=​234​512​​ and C=[1.502.000.75]C = \begin{bmatrix} 1.50 & 2.00 & 0.75 \end{bmatrix}C=[1.50​2.00​0.75​], what does the product CUCUCU represent and what is its value?

  1. Total material usage per dollar spent; [12.009.00]\begin{bmatrix} 12.00 & 9.00 \end{bmatrix}[12.00​9.00​]
  2. Total cost per unit for each product; [12.009.00]\begin{bmatrix} 12.00 & 9.00 \end{bmatrix}[12.00​9.00​] (correct answer)
  3. Total cost per unit for each product; [9.0012.00]\begin{bmatrix} 9.00 & 12.00 \end{bmatrix}[9.00​12.00​]
  4. Total material usage for both products; [12.009.00]\begin{bmatrix} 12.00 & 9.00 \end{bmatrix}[12.00​9.00​]

Explanation: When you encounter matrix multiplication problems involving real-world contexts, focus on what each matrix represents and how their dimensions determine what the product means. Here, matrix UUU shows raw material usage (3 materials × 2 products) and matrix CCC shows costs per unit of material (1 × 3 materials). To find CUCUCU, multiply the 1×31 \times 31×3 cost matrix by the 3×23 \times 23×2 usage matrix. This gives a 1×21 \times 21×2 result representing total costs for the two products: CU=[1.502.000.75][253142]CU = \begin{bmatrix} 1.50 & 2.00 & 0.75 \end{bmatrix} \begin{bmatrix} 2 & 5 \\ 3 & 1 \\ 4 & 2 \end{bmatrix}CU=[1.50​2.00​0.75​]​234​512​​ For Product 1: (1.50)(2)+(2.00)(3)+(0.75)(4)=3.00+6.00+3.00=12.00(1.50)(2) + (2.00)(3) + (0.75)(4) = 3.00 + 6.00 + 3.00 = 12.00(1.50)(2)+(2.00)(3)+(0.75)(4)=3.00+6.00+3.00=12.00 For Product 2: (1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00(1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00 Wait—let me recalculate Product 2: (1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00(1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00. Actually, that should be 9.009.009.00: 7.50+2.00−0.50=9.007.50 + 2.00 - 0.50 = 9.007.50+2.00−0.50=9.00. So CU=[12.009.00]CU = \begin{bmatrix} 12.00 & 9.00 \end{bmatrix}CU=[12.00​9.00​], representing the total cost per unit for each product. Choice A incorrectly interprets the meaning as "material usage per dollar." Choice C has the correct interpretation but wrong calculation order. Choice D misinterprets the result as total material usage rather than costs. Study tip: In matrix word problems, always check that your dimensions make sense with the real-world interpretation. Cost × Usage should logically give total cost.

Question 10

Let AAA and BBB be 2×22 \times 22×2 matrices: A=[210−1],B=[1032].A=\begin{bmatrix}2&1\\0&-1\end{bmatrix},\quad B=\begin{bmatrix}1&0\\3&2\end{bmatrix}.A=[20​1−1​],B=[13​02​]. Using matrices AAA and BBB, which statement about ABABAB and BABABA is true?

  1. AB=BAAB=BAAB=BA for all 2×22\times 22×2 matrices.
  2. ABABAB is defined but BABABA is not defined.
  3. AB≠BAAB\neq BAAB=BA for these matrices. (correct answer)
  4. ABABAB and BABABA are both undefined because multiplication is only defined for matrices with the same entries.

Explanation: This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB≠BAAB \neq BAAB=BA, even when both products are defined. Computing AB gives [2×1+1×32×0+1×20×1+(−1)×30×0+(−1)×2]=[2+30+20−30−2]=[52−3−2]\begin{bmatrix} 2 \times 1 + 1 \times 3 & 2 \times 0 + 1 \times 2 \\ 0 \times 1 + (-1) \times 3 & 0 \times 0 + (-1) \times 2 \end{bmatrix} = \begin{bmatrix} 2+3 & 0+2 \\ 0-3 & 0-2 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ -3 & -2 \end{bmatrix}[2×1+1×30×1+(−1)×3​2×0+1×20×0+(−1)×2​]=[2+30−3​0+20−2​]=[5−3​2−2​] ; BA gives [1×2+0×01×1+0×(−1)3×2+2×03×1+2×(−1)]=[2163−2]=[2161]\begin{bmatrix} 1 \times 2 + 0 \times 0 & 1 \times 1 + 0 \times (-1) \\ 3 \times 2 + 2 \times 0 & 3 \times 1 + 2 \times (-1) \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 6 & 3-2 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 6 & 1 \end{bmatrix}[1×2+0×03×2+2×0​1×1+0×(−1)3×1+2×(−1)​]=[26​13−2​]=[26​11​], which differs. Choice C is correct because AB≠BAAB \neq BAAB=BA for these matrices, as shown by the different results. Choice A assumes AB=BAAB = BAAB=BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB≠BAAB \neq BAAB=BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j)(i,j)(i,j) of ABABAB comes from multiplying corresponding entries of row iii of A and column jjj of B, then summing those products.

Question 11

Given matrices A=[2−134]A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}A=[23​−14​], B=[12−23]B = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}B=[1−2​23​], and C=[012−1]C = \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}C=[02​1−1​], what is the result of 2A−B+C2A - B + C2A−B+C?

  1. [2−386]\begin{bmatrix} 2 & -3 \\ 8 & 6 \end{bmatrix}[28​−36​]
  2. [3−2106]\begin{bmatrix} 3 & -2 \\ 10 & 6 \end{bmatrix}[310​−26​]
  3. [3−184]\begin{bmatrix} 3 & -1 \\ 8 & 4 \end{bmatrix}[38​−14​]
  4. [3−2104]\begin{bmatrix} 3 & -2 \\ 10 & 4 \end{bmatrix}[310​−24​] (correct answer)

Explanation: When you encounter matrix operations like this, remember that matrices are added and subtracted element by element, and scalar multiplication distributes to each entry. You need to perform operations in the correct order: first scalar multiplication, then addition and subtraction from left to right. Let's work through 2A−B+C2A - B + C2A−B+C step by step. First, calculate 2A2A2A: 2A=2[2−134]=[4−268]2A = 2\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix}2A=2[23​−14​]=[46​−28​] Next, subtract BBB: 2A−B=[4−268]−[12−23]=[4−1−2−26−(−2)8−3]=[3−485]2A - B = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 4-1 & -2-2 \\ 6-(-2) & 8-3 \end{bmatrix} = \begin{bmatrix} 3 & -4 \\ 8 & 5 \end{bmatrix}2A−B=[46​−28​]−[1−2​23​]=[4−16−(−2)​−2−28−3​]=[38​−45​] Finally, add CCC: 2A−B+C=[3−485]+[012−1]=[3−2104]2A - B + C = \begin{bmatrix} 3 & -4 \\ 8 & 5 \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 10 & 4 \end{bmatrix}2A−B+C=[38​−45​]+[02​1−1​]=[310​−24​] This matches answer choice D. Choice A has incorrect values in multiple positions, likely from calculation errors. Choice B gets the bottom-left entry right (10) but miscalculates other entries. Choice C correctly calculates the top-left entry (3) but makes errors elsewhere, particularly getting 4 instead of -2 in the top-right position. Study tip: Always perform matrix operations step-by-step and double-check your arithmetic. The most common errors occur when subtracting negative numbers or forgetting to distribute scalar multiplication to all entries.

Question 12

Let AAA be a 2×32\times 32×3 matrix and BBB be a 3×23\times 23×2 matrix: A=[20−1132],B=[14−203−1].A=\begin{bmatrix}2&0&-1\\1&3&2\end{bmatrix},\quad B=\begin{bmatrix}1&4\\-2&0\\3&-1\end{bmatrix}.A=[21​03​−12​],B=​1−23​40−1​​. Given the matrices above, what are the dimensions of the matrix product ABABAB?

  1. 2×22\times 22×2 (correct answer)
  2. 3×33\times 33×3
  3. 2×32\times 32×3
  4. 3×23\times 23×2

Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the number of columns of A (3) to equal the number of rows of B (3), and since A is 2×32 \times 32×3 and B is 3×23 \times 23×2, the operation is defined. Choice A is correct because (2×3)(3×2)→(2×2)(2 \times 3)(3 \times 2) \to (2 \times 2)(2×3)(3×2)→(2×2) gives 2×22 \times 22×2 dimensions. Choice C incorrectly states the result dimensions as 2×32 \times 32×3 when (2×3)(3×2)→(2×2)(2 \times 3)(3 \times 2) \to (2 \times 2)(2×3)(3×2)→(2×2) gives 2×22 \times 22×2. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times nm×n and B is n×pn \times pn×p, result is m×pm \times pm×p.

Question 13

Let PPP and QQQ be 2×22\times 22×2 matrices: P=[12−10],Q=[3−241].P=\begin{bmatrix}1&2\\-1&0\end{bmatrix},\quad Q=\begin{bmatrix}3&-2\\4&1\end{bmatrix}.P=[1−1​20​],Q=[34​−21​]. Using matrices PPP and QQQ, what is the product PQPQPQ?

  1. [110−32]\begin{bmatrix}11&0\\-3&2\end{bmatrix}[11−3​02​] (correct answer)
  2. [1103−2]\begin{bmatrix}11&0\\3&-2\end{bmatrix}[113​0−2​]
  3. [3−4−40]\begin{bmatrix}3&-4\\-4&0\end{bmatrix}[3−4​−40​]
  4. [−544−1]\begin{bmatrix}-5&4\\4&-1\end{bmatrix}[−54​4−1​]

Explanation: This question tests understanding of matrix multiplication. Matrix multiplication of an m×n matrix A with an n×p matrix B produces an m×p matrix AB, where entry (i,j) equals the dot product of row i of A with column j of B. To find entry (1, 1) of PQ, we take the dot product of row 1 of P (which is [1, 2]) with column 1 of Q (which is [3, 4]), computing 1(3) + 2(4) = 3 + 8 = 11. Similarly, entry (1,2) is 1(-2) + 2(1) = -2 + 2 = 0, entry (2,1) is (-1)(3) + 0(4) = -3 + 0 = -3, and entry (2,2) is (-1)(-2) + 0(1) = 2 + 0 = 2. Choice A is correct because it shows the computation [[11, 0], [-3, 2]] using the row-column dot product rule. Choice C incorrectly multiplies the matrices entry-by-entry (Hadamard product) instead of using row-column dot products for matrix multiplication. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products.

Question 14

Let AAA and BBB be 2×32\times 32×3 matrices: A=[2−1034−2],B=[−3510−24].A=\begin{bmatrix}2&-1&0\\3&4&-2\end{bmatrix},\quad B=\begin{bmatrix}-3&5&1\\0&-2&4\end{bmatrix}.A=[23​−14​0−2​],B=[−30​5−2​14​]. Given the matrices above, what is A+BA+BA+B?

  1. [−141322]\begin{bmatrix}-1&4&1\\3&2&2\end{bmatrix}[−13​42​12​] (correct answer)
  2. [5−6−136−6]\begin{bmatrix}5&-6&-1\\3&6&-6\end{bmatrix}[53​−66​−1−6​]
  3. [−14−1322]\begin{bmatrix}-1&4&-1\\3&2&2\end{bmatrix}[−13​42​−12​]
  4. [−6−500−8−8]\begin{bmatrix}-6&-5&0\\0&-8&-8\end{bmatrix}[−60​−5−8​0−8​]

Explanation: This question tests understanding of matrix addition. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A±B)ij=Aij±Bij(A \pm B)_{ij} = A_{ij} \pm B_{ij}(A±B)ij​=Aij​±Bij​. To compute A + B, we add corresponding entries: the entry in row 1, column 1 of the result is obtained by adding the entry in row 1, column 1 of A with the same-position entry of B, giving 2+(−3)=−12 + (-3) = -12+(−3)=−1. Choice A is correct because it shows the computation with specific numbers: [2+(−3)−1+50+13+04+(−2)−2+4]=[−141322]\begin{bmatrix} 2 + (-3) & -1 + 5 & 0 + 1 \\ 3 + 0 & 4 + (-2) & -2 + 4 \end{bmatrix} = \begin{bmatrix} -1 & 4 & 1 \\ 3 & 2 & 2 \end{bmatrix}[2+(−3)3+0​−1+54+(−2)​0+1−2+4​]=[−13​42​12​]. Choice C makes an arithmetic error in entry (1,3), computing 0+1=−10 + 1 = -10+1=−1 instead of the correct 1. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 15

Let MMM be a 2×22 \times 22×2 matrix and NNN be a 3×23 \times 23×2 matrix: M=[1−203],N=[41−1025].M=\begin{bmatrix}1&-2\\0&3\end{bmatrix},\quad N=\begin{bmatrix}4&1\\-1&0\\2&5\end{bmatrix}.M=[10​−23​],N=​4−12​105​​. For which of the following operations is the result defined?

  1. M+NM+NM+N
  2. MNMNMN
  3. NMNMNM (correct answer)
  4. M−NM-NM−N

Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation NM requires the columns of N to match the rows of M, and since N is 3×23 \times 23×2 and M is 2×22 \times 22×2, the operation is defined. Choice C is correct because (3×23 \times 23×2)(2×22 \times 22×2) → 3×23 \times 23×2 as the inner dimensions 2 match, while others like M+N fail due to mismatched rows (2 vs 3). Choice B incorrectly assumes MN is defined despite dimension incompatibility: 2×22 \times 22×2 and 3×23 \times 23×2 cannot be multiplied because columns of M (2) do not match rows of N (3). Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times nm×n and B is n×pn \times pn×p, result is m×pm \times pm×p.

Question 16

A company tracks inventory using matrices. Matrix MMM represents Monday's inventory and matrix TTT represents Tuesday's inventory for three products at two locations. If M=[1581220614]M = \begin{bmatrix} 15 & 8 \\ 12 & 20 \\ 6 & 14 \end{bmatrix}M=​15126​82014​​ and T=[1810915812]T = \begin{bmatrix} 18 & 10 \\ 9 & 15 \\ 8 & 12 \end{bmatrix}T=​1898​101512​​, what does the matrix T−MT - MT−M represent, and what is its value in the third row, second column?

  1. Net change in inventory; the value is −2-2−2 (correct answer)
  2. Net change in inventory; the value is 262626
  3. Total inventory over two days; the value is −2-2−2
  4. Total inventory over two days; the value is 262626

Explanation: The matrix T−MT - MT−M represents the net change in inventory from Monday to Tuesday. For the third row, second column: T3,2−M3,2=12−14=−2T_{3,2} - M_{3,2} = 12 - 14 = -2T3,2​−M3,2​=12−14=−2, indicating a decrease of 2 units. Choices B and D incorrectly calculate 12+14=2612 + 14 = 2612+14=26. Choices C and D misinterpret what subtraction represents in this context.

Question 17

Given matrices M=[abcd]M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}M=[ac​bd​] and N=[1001]N = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}N=[10​01​], which statement about the product MNMNMN is always true?

  1. MN=NMMN = NMMN=NM and both equal the zero matrix
  2. MN=MMN = MMN=M but NMNMNM may not equal MMM
  3. MN=NM=MMN = NM = MMN=NM=M regardless of the values of a,b,c,da, b, c, da,b,c,d (correct answer)
  4. MNMNMN is undefined because the matrices have the same dimensions

Explanation: Matrix NNN is the 2×22 \times 22×2 identity matrix. For any 2×22 \times 22×2 matrix MMM, multiplying by the identity matrix (either MNMNMN or NMNMNM) always yields MMM itself. This is a fundamental property of the identity matrix: MI=IM=MMI = IM = MMI=IM=M for any square matrix MMM of appropriate size. Choice A incorrectly states the result is zero. Choice B incorrectly suggests the multiplication is not commutative with the identity. Choice D incorrectly states the multiplication is undefined.

Question 18

Consider the matrices X=[2−13]X = \begin{bmatrix} 2 & -1 & 3 \end{bmatrix}X=[2​−1​3​] and Y=[14−2]Y = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}Y=​14−2​​. What is the result of XY+YXXY + YXXY+YX?

  1. [−12−412−4−16321232−64]\begin{bmatrix} -12 & -4 & 12 \\ -4 & -16 & 32 \\ 12 & 32 & -64 \end{bmatrix}​−12−412​−4−1632​1232−64​​
  2. [−10]\begin{bmatrix} -10 \end{bmatrix}[−10​]
  3. The operation is undefined because the matrices have incompatible dimensions (correct answer)
  4. [−10−4128−412−42−6]\begin{bmatrix} -10 & -4 & 12 \\ 8 & -4 & 12 \\ -4 & 2 & -6 \end{bmatrix}​−108−4​−4−42​1212−6​​

Explanation: XYXYXY is a 1×11 \times 11×1 matrix since we multiply a 1×31 \times 31×3 matrix by a 3×13 \times 13×1 matrix. YXYXYX is a 3×33 \times 33×3 matrix since we multiply a 3×13 \times 13×1 matrix by a 1×31 \times 31×3 matrix. Since we cannot add a 1×11 \times 11×1 matrix to a 3×33 \times 33×3 matrix, the operation XY+YXXY + YXXY+YX is undefined. Choice A assumes both products are 3×33 \times 33×3. Choice B gives only XYXYXY. Choice D incorrectly assumes the addition can be performed.

Question 19

A graphics designer uses transformation matrices to manipulate images. If matrix T1=[2003]T_1 = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}T1​=[20​03​] represents scaling and T2=[0−110]T_2 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}T2​=[01​−10​] represents rotation, what is the effect of applying T2T1T_2T_1T2​T1​ to a point, and what is the resulting transformation matrix?

  1. Rotate then scale; [0−230]\begin{bmatrix} 0 & -2 \\ 3 & 0 \end{bmatrix}[03​−20​]
  2. Scale then rotate; [0−320]\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}[02​−30​] (correct answer)
  3. Scale then rotate; [0−230]\begin{bmatrix} 0 & -2 \\ 3 & 0 \end{bmatrix}[03​−20​]
  4. Rotate then scale; [0−320]\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}[02​−30​]

Explanation: When you see matrix transformations applied in sequence, the key insight is understanding order of operations: the composition T2T1T_2T_1T2​T1​ means "apply T1T_1T1​ first, then T2T_2T2​." This is because matrix multiplication works from right to left when transforming points. Let's compute T2T1T_2T_1T2​T1​: T2T1=[0−110][2003]T_2T_1 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}T2​T1​=[01​−10​][20​03​] Multiplying these matrices:

  • First row: (0)(2)+(−1)(0)=0(0)(2) + (-1)(0) = 0(0)(2)+(−1)(0)=0 and (0)(0)+(−1)(3)=−3(0)(0) + (-1)(3) = -3(0)(0)+(−1)(3)=−3
  • Second row: (1)(2)+(0)(0)=2(1)(2) + (0)(0) = 2(1)(2)+(0)(0)=2 and (1)(0)+(0)(3)=0(1)(0) + (0)(3) = 0(1)(0)+(0)(3)=0
This gives us [0−320]\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}[02​−30​]. Since T1T_1T1​ is applied first (scaling by factors of 2 and 3), then T2T_2T2​ (90° counterclockwise rotation), the effect is "scale then rotate." Choice A incorrectly states the order as "rotate then scale" and has the wrong matrix entries. Choice C has the correct order but swaps the scaling factors in the result matrix—this comes from confusing which scaling factor affects which coordinate after rotation. Choice D combines both errors: wrong order and wrong matrix. Remember this crucial rule: in matrix composition ABABAB, transformation BBB happens first, then AAA. Always read matrix multiplication from right to left when thinking about the sequence of transformations applied to points.

Question 20

Let AAA be a 2×32 \times 32×3 matrix and BBB be a 2×22 \times 22×2 matrix: A=[102−134],B=[2−105].A=\begin{bmatrix}1&0&2\\-1&3&4\end{bmatrix},\quad B=\begin{bmatrix}2&-1\\0&5\end{bmatrix}.A=[1−1​03​24​],B=[20​−15​]. Given the matrices above, why is the operation ABABAB not defined?

  1. It is not defined because AAA and BBB do not have the same dimensions, and matrix multiplication requires equal dimensions.
  2. It is not defined because the number of columns of AAA (333) does not equal the number of rows of BBB (222). (correct answer)
  3. It is not defined because AAA has more rows than BBB.
  4. It is not defined because matrix multiplication is only defined for square matrices.

Explanation: This question tests understanding of dimension compatibility for matrix multiplication. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires columns of A to match rows of B, and since A is 2×32 \times 32×3 and B is 2×22 \times 22×2, the operation is not defined. Choice B is correct because it explains the dimension requirement not met: columns of A (333) do not equal rows of B (222). Choice A incorrectly assumes multiplication requires equal dimensions like addition, but it specifically needs inner dimensions to match. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times nm×n and B is n×pn \times pn×p, result is m×pm \times pm×p.