Matrix Addition, Subtraction, and Multiplication - Precalculus

1. A manufacturer produces two products using three raw materials. The usage matrix $U$ shows units of each raw material needed per product, and the cost matrix $C$ shows the cost per unit of each raw material. Given $U = \begin{bmatrix} 2 & 5 \\ 3 & 1 \\ 4 & 2 \end{bmatrix}$ and $C = \begin{bmatrix} 1.50 & 2.00 & 0.75 \end{bmatrix}$, what does the product $CU$ represent and what is its value?
   Answer: Total cost per unit for each product; $\begin{bmatrix} 12.00 & 9.00 \end{bmatrix}$ 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 $U$ shows raw material usage (3 materials × 2 products) and matrix $C$ shows costs per unit of material (1 × 3 materials). To find $CU$, multiply the $1 \times 3$ cost matrix by the $3 \times 2$ usage matrix. This gives a $1 \times 2$ result representing total costs for the two products: $CU = \begin{bmatrix} 1.50 & 2.00 & 0.75 \end{bmatrix} \begin{bmatrix} 2 & 5 \\ 3 & 1 \\ 4 & 2 \end{bmatrix}$ For Product 1: $(1.50)(2) + (2.00)(3) + (0.75)(4) = 3.00 + 6.00 + 3.00 = 12.00$ For Product 2: $(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00$ Wait—let me recalculate Product 2: $(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00$. Actually, that should be $9.00$: $7.50 + 2.00 - 0.50 = 9.00$. So $CU = \begin{bmatrix} 12.00 & 9.00 \end{bmatrix}$, 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.
2. Matrices $P$ and $Q$ are given below: $P=\begin{bmatrix}1&2\\-1&0\end{bmatrix},\quad Q=\begin{bmatrix}3&-2\\4&1\end{bmatrix}.$ Using matrices $P$ and $Q$, what is the product $PQ$?
   Answer: $\begin{bmatrix}11&0\\-3&2\end{bmatrix}$ 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. Choice A is correct because it shows the computation with specific numbers: [[1*3 + 2*4, 1*(-2) + 2*1],[-1*3 + 0*4, -1*(-2) + 0*1]] = [[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.
3. Let $A$ and $B$ be $2\times 2$ matrices: $A=\begin{bmatrix}-1&2\\3&0\end{bmatrix},\quad B=\begin{bmatrix}4&-1\\-2&5\end{bmatrix}.$ Given the matrices above, what is $2A-B$?
   Answer: $\begin{bmatrix}-6&5\\8&-5\end{bmatrix}$ This question tests understanding of matrix addition and subtraction with scalar multiplication. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. To compute 2A - B, first $2A = \begin{bmatrix} -2 & 4 \\ 6 & 0 \end{bmatrix}$, then subtract B: $\begin{bmatrix} -2-4 & 4-(-1) \\ 6-(-2) & 0-5 \end{bmatrix} = \begin{bmatrix} -6 & 5 \\ 8 & -5 \end{bmatrix}$. Choice A is correct because it shows the computation with specific numbers: 2A - B = $\begin{bmatrix} -2-4 & 4-(-1) \\ 6-(-2) & 0-5 \end{bmatrix} = \begin{bmatrix} -6 & 5 \\ 8 & -5 \end{bmatrix}$. Choice B has a sign error in the entry (2,1), likely from incorrectly subtracting as $6 - (-2) = 8$ but listing -8 instead of 8. 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.
4. Given matrices $A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$, and $C = \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}$, what is the result of $2A - B + C$?
   Answer: $\begin{bmatrix} 3 & -2 \\ 10 & 4 \end{bmatrix}$ 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 $2A - B + C$ step by step. First, calculate $2A$: $2A = 2\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix}$ Next, subtract $B$: $2A - B = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 4-1 & -2-2 \\ 6-(-2) & 8-3 \end{bmatrix} = \begin{bmatrix} 3 & -4 \\ 8 & 5 \end{bmatrix}$ Finally, add $C$: $2A - B + C = \begin{bmatrix} 3 & -4 \\ 8 & 5 \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 10 & 4 \end{bmatrix}$ 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.
5. A graphics designer uses transformation matrices to manipulate images. If matrix $T_1 = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ represents scaling and $T_2 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ represents rotation, what is the effect of applying $T_2T_1$ to a point, and what is the resulting transformation matrix?
   Answer: Scale then rotate; $\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}$ When you see matrix transformations applied in sequence, the key insight is understanding order of operations: the composition $T_2T_1$ means "apply $T_1$ first, then $T_2$." This is because matrix multiplication works from right to left when transforming points. Let's compute $T_2T_1$: $T_2T_1 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ Multiplying these matrices: - First row: $(0)(2) + (-1)(0) = 0$ and $(0)(0) + (-1)(3) = -3$ - Second row: $(1)(2) + (0)(0) = 2$ and $(1)(0) + (0)(3) = 0$ This gives us $\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}$. Since $T_1$ is applied first (scaling by factors of 2 and 3), then $T_2$ (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 $AB$, transformation $B$ happens first, then $A$. Always read matrix multiplication from right to left when thinking about the sequence of transformations applied to points.
6. Let $A$ and $B$ be $2\times 2$ matrices: $A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\quad B=\begin{bmatrix}0&1\\-1&2\end{bmatrix}.$ Given the matrices above, which statement about $AB$ and $BA$ is true?
   Answer: $AB\ne BA$ for these matrices. This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, $AB \neq BA$, even when both products are defined. Computing AB gives $\begin{bmatrix} -2 & 5 \\ -4 & 11 \end{bmatrix}$ while computing BA gives $\begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$, demonstrating that matrix multiplication depends on the order of the factors. Choice C is correct because it shows the reasoning with specific numbers: $AB \neq BA$ for these matrices. Choice A assumes $AB = BA$, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative ($AB \neq BA$ 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.
7. Let $A$ and $B$ be $2 \times 2$ matrices: $A=\begin{bmatrix}1&-2\\3&0\end{bmatrix},\quad B=\begin{bmatrix}4&1\\-1&2\end{bmatrix}.$ Based on the matrices defined, what is $2A-B$?
   Answer: $\begin{bmatrix}-2&-5\\7&-2\end{bmatrix}$ This question tests understanding of matrix addition, subtraction, and scalar multiplication. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. To compute $2A - B$, we first scale $A$ by 2 to get $\begin{bmatrix} 2 & -4 \\ 6 & 0 \end{bmatrix}$, then subtract $B$ entry-by-entry, giving for example in row 1 column 1: $2 - 4 = -2$. Choice A is correct because it shows the computation with specific numbers: $\begin{bmatrix} -2 & -5 \\ 7 & -2 \end{bmatrix}$ from operations like $6 - (-1) = 7$. Choice D has a sign error in the entry (2,2), computing 2 instead of -2, likely from adding instead of subtracting. 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.
8. Let $A$ and $B$ be matrices with dimensions $A$ is $2 \times 3$ and $B$ is $2 \times 2$: $A=\begin{bmatrix}1&-2&0\\3&1&4\end{bmatrix},\quad B=\begin{bmatrix}2&1\\0&-3\end{bmatrix}.$ Given the matrices above, why is the operation $AB$ not defined?
   Answer: It is not defined because the number of columns of $A$ ($3$) does not equal the number of rows of $B$ ($2$). 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 ($2$), and since A is $2 \times 3$ and B is $2 \times 2$, the operation is not defined. Choice B is correct because it states the dimension requirement not met: columns of A ($3$) $\neq$ rows of B ($2$). Choice A incorrectly assumes the operation is defined despite dimension incompatibility: $2 \times 3$ and $2 \times 2$ cannot be multiplied because inner dimensions $3 \neq 2$. 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 $m \times n$ and B is $n \times p$, result is $m \times p$.
9. Let $M$ be a $2 \times 2$ matrix and $N$ be a $3 \times 2$ matrix: $M=\begin{bmatrix}1&-2\\0&3\end{bmatrix},\quad N=\begin{bmatrix}4&1\\-1&0\\2&5\end{bmatrix}.$ For which of the following operations is the result defined?
   Answer: $NM$ 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 $3 \times 2$ and M is $2 \times 2$, the operation is defined. Choice C is correct because ($3 \times 2$)($2 \times 2$) → $3 \times 2$ 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: $2 \times 2$ and $3 \times 2$ 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 $m \times n$ and B is $n \times p$, result is $m \times p$.
10. Let $A$ and $B$ be $2\times 3$ matrices: $A=\begin{bmatrix}2&-1&0\\3&4&-2\end{bmatrix},\quad B=\begin{bmatrix}-3&5&1\\0&-2&4\end{bmatrix}.$ Given the matrices above, what is $A+B$?
    Answer: $\begin{bmatrix}-1&4&1\\3&2&2\end{bmatrix}$ 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: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. 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 $2 + (-3) = -1$. Choice A is correct because it shows the computation with specific numbers: $\begin{bmatrix} -1 & 4 & 1 \\ 3 & 2 & 2 \end{bmatrix}$ from adding each pair like $-1 + 5 = 4$ and $-2 + 4 = 2$. Choice B makes an arithmetic error by subtracting instead of adding, computing A - B to get entries like $2 - (-3) = 5$ and $-2 - 4 = -6$. 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.
11. Let $M$ and $N$ be $2 \times 2$ matrices: $M=\begin{bmatrix}2&-1\\3&4\end{bmatrix},\quad N=\begin{bmatrix}0&5\\-2&1\end{bmatrix}.$ Using matrices $M$ and $N$, what is $2M-N$?
    Answer: $\begin{bmatrix}4&-7\\8&7\end{bmatrix}$ This question tests understanding of matrix addition-subtraction and multiplication. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. To compute 2M - N, we subtract corresponding entries after scaling: the entry in row 1, column 1 of the result is obtained by subtracting the entry in row 1, column 1 of N from the same-position entry of 2M, giving $4 - 0 = 4$. Choice B is correct because it shows the computation with specific numbers: $\begin{bmatrix} 4 - 0 & -2 - 5 \\ 6 - (-2) & 8 - 1 \end{bmatrix} = \begin{bmatrix} 4 & -7 \\ 8 & 7 \end{bmatrix}$. Choice A has a sign error in the entry (1,2), likely from incorrectly subtracting where should subtract correctly. 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.
12. Given matrices $M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $N = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, which statement about the product $MN$ is always true?
    Answer: $MN = NM = M$ regardless of the values of $a, b, c, d$ Matrix $N$ is the $2 \times 2$ identity matrix. For any $2 \times 2$ matrix $M$, multiplying by the identity matrix (either $MN$ or $NM$) always yields $M$ itself. This is a fundamental property of the identity matrix: $MI = IM = M$ for any square matrix $M$ 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.
13. Let $A$ and $B$ be $2\times 2$ matrices: $A=\begin{bmatrix}1&2\\0&1\end{bmatrix},\quad B=\begin{bmatrix}2&0\\1&3\end{bmatrix}.$ Which statement about the products $AB$ and $BA$ is true?
    Answer: $AB\ne BA$ 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. Since A is 2×2 and B is 2×2, both products AB and BA are defined (inner dimensions match). Computing AB: entry (1,1) = 1(2) + 2(1) = 4, entry (1,2) = 1(0) + 2(3) = 6, entry (2,1) = 0(2) + 1(1) = 1, entry (2,2) = 0(0) + 1(3) = 3, giving AB = [[4, 6], [1, 3]]. Computing BA: entry (1,1) = 2(1) + 0(0) = 2, entry (1,2) = 2(2) + 0(1) = 4, entry (2,1) = 1(1) + 3(0) = 1, entry (2,2) = 1(2) + 3(1) = 5, giving BA = [[2, 4], [1, 5]]. Since AB = [[4, 6], [1, 3]] and BA = [[2, 4], [1, 5]], we have AB ≠ BA. Choice C is correct because it correctly states that AB ≠ BA, demonstrating that matrix multiplication depends on the order of the factors. 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.
14. Let $A$ and $B$ be $2\times 3$ matrices: $A=\begin{bmatrix}2&-1&0\\3&4&-2\end{bmatrix},\quad B=\begin{bmatrix}-3&5&1\\0&-2&4\end{bmatrix}.$ Given the matrices above, what is $A+B$?
    Answer: $\begin{bmatrix}-1&4&1\\3&2&2\end{bmatrix}$ 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: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. 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 $2 + (-3) = -1$. Choice A is correct because it shows the computation with specific numbers: $\begin{bmatrix} 2 + (-3) & -1 + 5 & 0 + 1 \\ 3 + 0 & 4 + (-2) & -2 + 4 \end{bmatrix} = \begin{bmatrix} -1 & 4 & 1 \\ 3 & 2 & 2 \end{bmatrix}$. Choice C makes an arithmetic error in entry (1,3), computing $0 + 1 = -1$ 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.
15. Consider the matrices $X = \begin{bmatrix} 2 & -1 & 3 \end{bmatrix}$ and $Y = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}$. What is the result of $XY + YX$?
    Answer: The operation is undefined because the matrices have incompatible dimensions $XY$ is a $1 \times 1$ matrix since we multiply a $1 \times 3$ matrix by a $3 \times 1$ matrix. $YX$ is a $3 \times 3$ matrix since we multiply a $3 \times 1$ matrix by a $1 \times 3$ matrix. Since we cannot add a $1 \times 1$ matrix to a $3 \times 3$ matrix, the operation $XY + YX$ is undefined. Choice A assumes both products are $3 \times 3$. Choice B gives only $XY$. Choice D incorrectly assumes the addition can be performed.
16. A company tracks inventory using matrices. Matrix $M$ represents Monday's inventory and matrix $T$ represents Tuesday's inventory for three products at two locations. If $M = \begin{bmatrix} 15 & 8 \\ 12 & 20 \\ 6 & 14 \end{bmatrix}$ and $T = \begin{bmatrix} 18 & 10 \\ 9 & 15 \\ 8 & 12 \end{bmatrix}$, what does the matrix $T - M$ represent, and what is its value in the third row, second column?
    Answer: Net change in inventory; the value is $-2$ The matrix $T - M$ represents the net change in inventory from Monday to Tuesday. For the third row, second column: $T_{3,2} - M_{3,2} = 12 - 14 = -2$, indicating a decrease of 2 units. Choices B and D incorrectly calculate $12 + 14 = 26$. Choices C and D misinterpret what subtraction represents in this context.
17. Let $C$ and $D$ be $2\times 3$ matrices: $C=\begin{bmatrix}1&0&-2\\3&-1&4\end{bmatrix},\quad D=\begin{bmatrix}-3&2&5\\0&-4&1\end{bmatrix}.$ Based on the matrices defined, what is $C-D$?
    Answer: $\begin{bmatrix}4&-2&-7\\3&3&3\end{bmatrix}$ 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.
18. Let $A$ be a $2 \times 2$ matrix and $B$ be a $2 \times 2$ matrix: $A=\begin{bmatrix}2&1\\0&-3\end{bmatrix},\quad B=\begin{bmatrix}-1&4\\5&0\end{bmatrix}.$ Using matrices $A$ and $B$, what is the entry in row $2$, column $1$ of the matrix product $AB$? (Rows and columns are 1-indexed.)
    Answer: $-15$ This question tests understanding of matrix multiplication. Matrix multiplication of an $m \times n$ matrix A with an $n \times p$ matrix B produces an $m \times p$ 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 $[0, -3]$) with column 1 of B (which is $[-1, 5]$), computing $0*(-1) + (-3)*5 = -15$. 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.
19. Let $C$ and $D$ be $3\times 2$ matrices: $C=\begin{bmatrix}1&-4\\0&2\\5&-1\end{bmatrix},\quad D=\begin{bmatrix}-2&3\\1&0\\-5&4\end{bmatrix}.$ Based on the matrices defined, what is $C-D$?
    Answer: $\begin{bmatrix}3&-7\\-1&2\\10&-5\end{bmatrix}$ 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: $(A \pm B)_{ij} = A_{ij} \pm B_{ij}$. 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 $1 - (-2) = 3$. Choice B is correct because it shows the computation with specific numbers: $\begin{bmatrix} 1 - (-2) & -4 - 3 \\ 0 - 1 & 2 - 0 \\ 5 - (-5) & -1 - 4 \end{bmatrix} = \begin{bmatrix} 3 & -7 \\ -1 & 2 \\ 10 & -5 \end{bmatrix}$. Choice D incorrectly reverses the subtraction order, computing D - C instead of C - D, which gives a different result with opposite signs. 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.
20. Let $A$ and $B$ be $2\times 2$ matrices: $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix},\quad B=\begin{bmatrix}-4&5\\1&-2\end{bmatrix}.$ Given the matrices above, what is $A+B$?
    Answer: $\begin{bmatrix}-2&4\\1&1\end{bmatrix}$ 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.

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