(a) 𝐀 𝐁=( -4+4 6-12 -3+8 -20+10 30-30 -15+20 -32+12 48-36

step 1 In this problem, we're being asked to evaluate both expressions, and they look alike. However, t

(a) 𝐀 𝐁=( -4+4 6-12 -3+8 -20+10 30-30 ...

(a) $\mathbf{A B}=\left(\begin{array}{rrr}-4+4 & 6-12 & -3+8 \\ -20+10 & 30-30 & -15+20 \\ -32+12 & 48-36 & -24+24\end{array}\right)=\left(\begin{array}{rrr}0 & -6 & 5 \\ -10 & 0 & 5 \\ -20 & 12 & 0\end{array}\right)$ (b) $\mathbf{B A}=\left(\begin{array}{rr}-4+30-24 & -16+60-36 \\ 1-15+16 & 4-30+24\end{array}\right)=\left(\begin{array}{rr}2 & 8 \\ 2 & -2\end{array}\right)$

(a) $\mathbf{A B}=\left(\begin{array}{rrr}-4+4 & 6-12 & -3+8 \\ -20+10 & 30-30 & -15+20 \\ -32+12 & 48-36 & -24+24\end{array}\right)=\left(\begin{array}{rrr}0 & -6 & 5 \\ -10 & 0 & 5 \\ -20 & 12 & 0\end{array}\right)$
(b) $\mathbf{B A}=\left(\begin{array}{rr}-4+30-24 & -16+60-36 \\ 1-15+16 & 4-30+24\end{array}\right)=\left(\begin{array}{rr}2 & 8 \\ 2 & -2\end{array}\right)$

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Key Concepts

Matrix Multiplication

Matrix multiplication is an operation where two matrices are combined to produce a third matrix by taking the dot product of rows and columns. For each element in the resulting matrix, you multiply corresponding entries from a row of the first matrix and a column of the second matrix and then sum these products. This process requires that the number of columns in the first matrix is equal to the number of rows in the second matrix.

Non-Commutativity of Matrix Multiplication

Matrix multiplication is non-commutative, meaning that the order in which two matrices are multiplied affects the result. As illustrated, the product AB can be different from the product BA, not only in the values of the resulting entries but also in the dimensions of the resulting matrices.

Dimension Compatibility

For matrix multiplication to be defined, the dimensions must be compatible; specifically, the number of columns in the first matrix must equal the number of rows in the second matrix. This rule also determines the dimensions of the resultant matrix, which, in general, will have the number of rows of the first matrix and the number of columns of the second matrix.

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