Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A. A=[010001100],B=[001100010]A = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}

Accepted Answer

Step 1: Identify the matrices A and B. Matrix A is given as $$A = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{bmatrix}$$ and matrix B is $$B = \begin{bmatrix} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}. $$Step 2: To find the product $$AB$$, multiply matrix A by matrix B. Recall that the element in the $$i^{th}$$ row and $$j^{th}$$ column of the product matrix is found by taking the dot product of the $$i^{th}$$ row of A with the $$j^{th}$$ column of B. Formally, $$(AB)_{ij} = \sum_{k=1}^3 A_{ik} B_{kj}. $$Step 3: Perform the multiplication for each element of the product matrix $$AB by $$calculating the dot products row by row and column by column. Step 4: Repeat the process to find the product $$BA by $$multiplying matrix B by matrix A, using the same dot product method for each element: $$(BA)_{ij} = \sum_{k=1}^3 B_{ik} A_{kj}. $$Step 5: After finding both products $$AB$$ and $$BA$$, compare each with the identity matrix $$I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}. If $$both $$AB = I$$ and $$BA = I$$, then matrix B is the multiplicative inverse of matrix A.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}],$

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Matrix Multiplication

Multiplicative Inverse of a Matrix

Identity Matrix

Find the products AB and BA to determine whether B is the multiplicative inverse of A. A=[010001100],B=[001100010]A = \(\begin{bmatrix}\) 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \(\end{bmatrix}\)