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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 19
Chapter 6, Problem 19

Find the inverse, if it exists, for each matrix. [−1−234]\(\left\)[ \(\begin{matrix}\) -1 & -2 \\ 3 & 4 \(\end{matrix}\) \(\right\)]

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1
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 is nonzero. If \( \det(A) \neq 0 \), the inverse exists; otherwise, the inverse does not exist.
If the inverse exists, use the formula for the inverse of a 2x2 matrix: \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Substitute the values of \( a, b, c, d \) and the determinant into the formula to express the inverse matrix.

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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. This scalar value determines whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides insight into the matrix's scaling effect on area.
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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, providing a straightforward method to find the inverse.
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Related Practice
Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

y=6x+x2y=6x+x^2

4x−y=−34x-y=-3

Textbook Question

Write the system of equations associated with each augmented matrix . Do not solve.

1
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Textbook Question

Find the values of the variables for which each statement is true, if possible.

[05x−13y+241z]=[0w6−130418]\(\left\)[ \(\begin{matrix}\) 0 & 5 & x \\ -1 & 3 & y + 2 \\ 4 & 1 & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 0 & w & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \(\end{matrix}\) \(\right\)]

7
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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x3 + 4)/(9x3 - 4x)

3
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Textbook Question

Solve each system by elimination. In systems with fractions, first clear denominators.

4x + y = -23

x - 2y = -17

2
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Textbook Question

Find the inverse, if it exists, for each matrix. [510−3−6]\(\left\)[ \(\begin{matrix}\) 5 & 10 \\ -3 & -6 \(\end{matrix}\) \(\right\)]