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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 9
Chapter 6, Problem 9

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 \(\end{matrix}\) \(\right\)]

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Solve each system by substitution.

x - 5y = 8

x = 6y

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Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

[8−2463]\(\left\)[ \(\begin{matrix}\) 8 & -2 & 4 & 6 & 3 \(\end{matrix}\) \(\right\)]

Textbook Question

Evaluate each determinant.

∣93−3−1∣\(\left\)| \(\begin{matrix}\) 9 & 3 \\ -3 & -1 \(\end{matrix}\) \(\right\)|

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Textbook Question

Use the given row transformation to change each matrix as indicated.

[156−4−12371],4×row 1 added to row 2\(\left\)[ \(\begin{matrix}\) 1 & 5 & 6 \\ -4 & -1 & 2 \\ 3 & 7 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) 4 \(\times\) \(\text{row 1 added to row 2}\)

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Textbook Question

Evaluate each determinant.

∣−1−253∣\(\left\)| \(\begin{matrix}\) -1 & -2 \\ 5 & 3 \(\end{matrix}\) \(\right\)|

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x + 2)/((x + 2)(2x - 1))

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