Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[0−5−42],B=[3−1−54]A = \left[ \begin{matrix} 0 & -5 \ -4 & 2 \end{matrix} ight], \quad B = \left[ \begin{matrix} 3 & -1 \ -5 & 4 \end{matrix} ight]

Accepted Answer

First, identify the dimensions of matrices A and B. Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix. To find the product AB, multiply each row of matrix A by each column of matrix B. For each element in the resulting matrix, calculate the sum of the products of corresponding elements from the row of A and the column of B. Use the formula: $$(AB)_{ij} = \sum_{k} A_{ik} B_{kj}$$ where $$i is $$the row index and $$j is $$the column index. To find the product BA, repeat the process by multiplying each row of matrix B by each column of matrix A, again ensuring the dimensions are compatible. Use the same summation formula for each element of BA. If the dimensions of A and B do not allow multiplication in one order (for example, if the number of columns in B does not equal the number of rows in A), then that product (BA or AB) is undefined. After computing the sums for each element, write down the resulting matrices AB and BA. Remember that matrix multiplication is generally not commutative, so AB and BA may be different or one may be undefined.

For each pair of matrices A and B, find (a) AB and (b) BA. $A = [\begin{matrix} 0 & - 5 \\ - 4 & 2 \end{matrix}],$

Verified video answer for a similar problem:

Key Concepts

Matrix Multiplication

Order of Multiplication

Dimension Compatibility

For each pair of matrices A and B, find (a) AB and (b) BA. A=[0−5−42],B=[3−1−54]A = \(\left\)[ \(\begin{matrix}\) 0 & -5 \\ -4 & 2 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 3 & -1 \\ -5 & 4 \(\end{matrix}\) \(\right\)]