Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A. ﻿A=[4−3−54],B=[4354]A = \begin{bmatrix}
4 & -3 \
-5 & 4
\end{bmatrix}
, B = \begin{bmatrix}
4 & 3 \
5 & 4
\end{bmatrix}A=[4−5​−34​],B=[45​34​]

Accepted Answer

Step 1: Write down the matrices A and B clearly. Matrix A is given by $$A = \begin{bmatrix} 4 & -3 \ -5 & 4 \end{bmatrix}$$ and matrix B is $$B = \begin{bmatrix} 4 & 3 \ 5 & 4 \end{bmatrix}. $$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. So, for $$AB$$, calculate each element as follows: $$(AB)_{11} = 4 	imes 4 + (-3) 	imes 5,$$ $$(AB)_{12} = 4 	imes 3 + (-3) 	imes 4,$$ $$(AB)_{21} = (-5) 	imes 4 + 4 	imes 5,$$ $$(AB)_{22} = (-5) 	imes 3 + 4 	imes 4.$$ Step 3: Similarly, find the product $$BA by $$multiplying matrix B by matrix A. Use the same method of dot products for each element: $$(BA)_{11} = 4 	imes 4 + 3 	imes (-5),$$ $$(BA)_{12} = 4 	imes (-3) + 3 	imes 4,$$ $$(BA)_{21} = 5 	imes 4 + 4 	imes (-5),$$ $$(BA)_{22} = 5 	imes (-3) + 4 	imes 4.$$ Step 4: After computing both $$AB$$ and $$BA$$, compare each product to the identity matrix $$I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}. If $$both $$AB = I$$ and $$BA = I$$, then matrix B is the multiplicative inverse of matrix A. Step 5: Conclude by verifying whether both products equal the identity matrix. If yes, B is the inverse of A; if not, B is not the inverse.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A = \(\begin{bmatrix}\)4 & -3 \\-5 & 4\(\end{bmatrix}\), B = \(\begin{bmatrix}\)4 & 3 \\5 & 4\(\end{bmatrix}\)$

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Key Concepts

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=[4−3−54],B=[4354]A = \(\begin{bmatrix}\)4 & -3 \\-5 & 4\(\end{bmatrix}\), B = \(\begin{bmatrix}\)4 & 3 \\5 & 4\(\end{bmatrix}\)A=[4−5−34],B=[4534]