Use the fact that if A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, then A−1=1ad−bc[d−b−ca]$$A^{-1}$$ = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} to find the inverse of each matrix, if possible. Check that AA−1=$$I2AA^{-1}$$ = $$I_2$$ and A−1A=$$I2A^{-1} A$$ = $$I_2.A$$=[3−1−42]A = \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}A=[3−4−12]

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 15

Chapter 7, Problem 15

Use the fact that if $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ to find the inverse of each matrix, if possible. Check that $A A^{- 1} = I_{2}$ and $A^{- 1} A = I_{2}$ .
$A = $\begin{bmatrix}$3 & -1 \\-4 & 2$\end{bmatrix}$$

Verified step by step guidance

Identify the elements of matrix $A$: $a = 3$, $b = -1$, $c = -4$, and $d = 2$.

Calculate the determinant of $A$ using the formula $ad - bc$: compute $3 \times 2 - (-1) \times (-4)$.

Check if the determinant is non-zero. If it is zero, the inverse does not exist; if non-zero, proceed to find the inverse.

Use the inverse formula $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ and substitute the values of $a$, $b$, $c$, and $d$.

Multiply the scalar $\frac{1}{ad - bc}$ by the matrix $\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ to write the inverse matrix explicitly.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

10m

Was this helpful?

Key Concepts

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

Matrix Inverse

The inverse of a square matrix A, denoted A⁻¹, is a matrix that when multiplied by A yields the identity matrix I. For a 2x2 matrix, the inverse exists only if the determinant (ad - bc) is nonzero. The formula for the inverse involves swapping elements a and d, changing signs of b and c, and dividing by the determinant.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix A = [[a, b], [c, d]] is calculated as ad - bc. It is a scalar value that 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 factor and orientation.

Identity Matrix and Matrix Multiplication

The identity matrix I₂ is a 2x2 matrix with ones on the diagonal and zeros elsewhere. Multiplying any 2x2 matrix A by I₂ leaves A unchanged. To verify an inverse, multiply A by A⁻¹ and A⁻¹ by A; both products should equal I₂, confirming the correctness of the inverse.

Recommended video:

03:42

Finding Zeros & Their Multiplicity

Related Practice

Textbook Question

In Exercises 9 - 16, find the following matrices: c. - 4A

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}\)4x - 5y = 17 \\2x + 3y = 3\(\end{cases}$

Textbook Question

Perform each matrix row operation and write the new matrix.

$$\begin{bmatrix}$1 & -3 & 2 & $\vert$ & 0 \\3 & 1 & -1 & $\vert$ & 7 \\2 & -2 & 1 & $\vert$ & 3$\end{bmatrix}\]\quad$ -3R_1 + R_2$

views

Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\)2x + y - z = 2 \\3x + 3y - 2z = 3\(\end{cases}$

views

Textbook Question

In Exercises 9 - 16, find the following matrices: b. A - B

Textbook Question

In Exercises 9 - 16, find the following matrices: a. A + B