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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 83
Chapter 6, Problem 83

Find the inverse, if it exists, for each matrix.
[2153]\(\left\)[ \(\begin{matrix}\) 2 & 1 \\ 5 & 3 \(\end{matrix}\) \(\right\)]

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Identify the given 2x2 matrix as \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
Calculate the determinant of matrix \( A \) using the formula \( \det(A) = ad - bc \).
Check if the determinant \( \det(A) \) is nonzero. If it is zero, the inverse does not exist.
If the determinant is nonzero, find the inverse matrix using the formula \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Write the inverse matrix explicitly by substituting the values of \( a, b, c, d \) and the determinant into the formula.

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Key Concepts

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

Matrix Inverse

The inverse of a matrix A is another matrix, denoted A⁻¹, such that when multiplied together, they yield the identity matrix. Only square matrices with nonzero determinants have inverses. Finding the inverse is essential for solving matrix equations and understanding linear transformations.
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Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. It indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides information about the matrix's scaling and orientation effects.
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Determinants of 2×2 Matrices

Formula for the Inverse of a 2x2 Matrix

For a 2x2 matrix [[a, b], [c, d]] with nonzero determinant, the inverse is (1/det) times the matrix [[d, -b], [-c, a]]. This formula swaps the diagonal elements, changes the signs of the off-diagonal elements, and scales by the reciprocal of the determinant.
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Related Practice
Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA.

A=[−101011−1−10],B=[001010100]A = \(\left\)[ \(\begin{matrix}\) -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 10y

Textbook Question

Find AB and BA for the following matrices.

A=[abcd]andB=[1001]A = \(\left\)[ \(\begin{matrix}\) a & b \\ c & d \(\end{matrix}\) \(\right\)] \(\quad\) \(\text{and}\) \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 \\ 0 & 1 \(\end{matrix}\) \(\right\)]

Matrix B acts as the multiplicative element for 2 ×\(\times\) 2 square matrices.

Textbook Question

Perform each operation, if possible.

[−2550143−4−1][10−1−10011−1]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

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Textbook Question

Find the inverse, if it exists, for each matrix.

[2−101011−20]\(\left\)[ \(\begin{matrix}\) 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & -2 & 0 \(\end{matrix}\) \(\right\)]