Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A. A=[00−21−101101−10100−1],B=[1203011101011202]A = \begin{bmatrix} 0 & 0 & -2 & 1 \ -1 & 0 & 1 & 1 \ 0 & 1 & -1 & 0 \ 1 & 0 & 0 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 0 & 3 \ 0 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 \ 1 & 2 & 0 & 2 \end{bmatrix}

Accepted Answer

Step 1: Identify the matrices A and B as given:

$$A = \begin{bmatrix} 0 & 0 & -2 & 1 \ -1 & 0 & 1 & 1 \ 0 & 1 & 0 & 0 \ 1 & 0 & 0 & -1 \end{bmatrix}$$

$$B = \begin{bmatrix} 1 & 2 & 0 & 3 \ 0 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 \ 1 & 2 & 0 & 2 \end{bmatrix}$$ Step 2: Compute the product $$AB by $$multiplying matrix A by matrix B. Recall that the element in the $$i^{th}$$ row and $$j^{th}$$ column of $$AB is $$found by taking the dot product of the $$i^{th}$$ row of A with the $$j^{th}$$ column of B:

$$ (AB)_{ij} = \sum_{k=1}^4 A_{ik} 	imes B_{kj} $$ Step 3: Compute the product $$BA by $$multiplying matrix B by matrix A. Similarly, the element in the $$i^{th}$$ row and $$j^{th}$$ column of $$BA is $$found by the dot product of the $$i^{th}$$ row of B with the $$j^{th}$$ column of A:

$$ (BA)_{ij} = \sum_{k=1}^4 B_{ik} 	imes A_{kj} $$ Step 4: After calculating both $$AB$$ and $$BA$$, compare each product to the identity matrix $$I_4 of $$size 4x4:

$$ I_4$ = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} $$ Step 5: If both $$AB = I_4$ and BA$$ = $$I_4$$, then matrix B is the multiplicative inverse of matrix A. Otherwise, B is not the inverse of A.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A = [\begin{matrix} 0 & 0 & - 2 & 1 \\ - 1 & 0 & 1 & 1 \\ 0 & 1 & - 1 & 0 \\ 1 & 0 & 0 & - 1 \end{matrix}],$

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Key Concepts

Matrix Multiplication

Multiplicative Inverse of a Matrix

Identity Matrix