Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix In.) [5723]and[3−7−25]\left[ \begin{matrix} 5 & 7 \\ 2 & 3 \end{matrix} \right] \quad \text{and} \quad \left[ \begin{matrix} 3 & -7 \\ -2 & 5 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 9

6장, 문제 9

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I_n.)
$[\begin{matrix} 5 & 7 \\ 2 & 3 \end{matrix}]$

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Recall that two matrices $ A $ and $ B $ are inverses of each other if and only if their product in both orders equals the identity matrix $ I_n $. That is, $ AB = I_n $ and $ BA = I_n $.

Write down the two given matrices $ A $ and $ B $ explicitly, so you can perform matrix multiplication.

Calculate the product $ AB $ by multiplying matrix $ A $ by matrix $ B $. For 2x2 matrices, use the formula: $ (AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} $ for each element.

Calculate the product $ BA $ by multiplying matrix $ B $ by matrix $ A $ using the same method as above.

Compare both products $ AB $ and $ BA $ to the 2x2 identity matrix $ I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $. If both equal $ I_2 $, then $ A $ and $ B $ are inverses; otherwise, they are not.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Matrix Multiplication

Matrix multiplication involves combining two matrices by multiplying rows of the first matrix by columns of the second. For two 2x2 matrices, each element of the product is found by summing the products of corresponding entries. Understanding this operation is essential to verify if the product equals the identity matrix.

Identity Matrix

The identity matrix, denoted Iₙ for an n×n matrix, has 1s on the main diagonal and 0s elsewhere. Multiplying any matrix by the identity matrix leaves it unchanged. Checking if the product of two matrices equals the identity matrix helps determine if they are inverses.

Matrix Inverse

A matrix inverse is a matrix that, when multiplied by the original matrix, yields the identity matrix. Only square matrices with nonzero determinants have inverses. Verifying that the product of two matrices is the identity matrix confirms they are inverses of each other.