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.

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.

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.

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 $m \times p$.

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.

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 $2 \times 3$ and B is $2 \times 2$, the operation is not defined. Choice B is correct because it explains the dimension requirement not met: columns of A ($3$) do not equal rows of B ($2$). 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 $m \times n$ and B is $n \times p$, result is $m \times p$.

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 columns of A to match the rows of B, and since A is $2 \times 3$ and B is $3 \times 2$, the operation is defined. Choice A is correct because ($2 \times 3$)($3 \times 2$) $\rightarrow$ $2 \times 2$ as the inner dimensions 3 match, giving result dimensions $2 \times 2$. Choice D incorrectly assumes the operation is not defined despite dimension compatibility: $2 \times 3$ and $3 \times 2$ can be multiplied because the inner dimensions 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 $m \times p$. 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.

This question tests understanding of the non-commutative nature of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, $AB \ne BA$, even when both products are defined. Computing $AB$ gives $$\begin{bmatrix} 3 & 12 \\ 3 & -10 \end{bmatrix}$$ while computing $BA$ gives $$\begin{bmatrix} -1 & 8 \\ 9 & -6 \end{bmatrix}$$, demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because it shows the reasoning with specific computations proving $AB \ne BA$ 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 ($AB \ne BA$ 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.

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.

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.