Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \left[ \begin{matrix} 0 & 1 & -1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight], \quad B = \left[ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \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. Specifically, the element in the i-th row and j-th column of AB is found by summing the products of corresponding elements from the i-th row of A and the j-th column of B, using the formula: $$(AB)_{ij} = \sum_{k} A_{ik} B_{kj}. $$Write out the resulting matrix AB by calculating each element using the above formula, ensuring you perform the multiplication and addition carefully for each element. Next, to find the product BA, check if the multiplication is defined by confirming the number of columns in B equals the number of rows in A. If defined, repeat the multiplication process: multiply each row of B by each column of A using the same summation formula. Finally, write out the resulting matrix BA by calculating each element as before. Remember that matrix multiplication is 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. $A = [\begin{matrix} 0 & 1 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}],$

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Key Concepts

Matrix Multiplication

Order of Multiplication in Matrices

Dimension Compatibility

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \(\left\)[ \(\begin{matrix}\) 0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)]