Chapter 12: Q52. (page 637)

Find the inverse of each matrix, if it exists.
$[\begin{matrix} 4 & - 5 \\ 2 & - 1 \end{matrix}]$

Short Answer

Expert verified

The inverse of the given matrix is $[\begin{matrix} - \frac{1}{6} & \frac{5}{6} \\ - \frac{1}{3} & \frac{2}{3} \end{matrix}]$ .

Step by step solution

Step 1. Given Information.

Given, to find the inverse of the matrix $[\begin{matrix} 4 & - 5 \\ 2 & - 1 \end{matrix}]$ .

Step 2. Calculation.

The inverse of matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ is $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ , where $a d - b c \neq 0$ :

Here, $a = 4, b = - 5, c = 2, d = - 1$ .

Checking if the inverse exists:

$\begin{array}{l} a d - b c = (4) (- 1) - (- 5) (2) \\ a d - b c = - 4 + 10 \\ a d - b c = 6 \end{array}$

Hence, the inverse exists.

The inverse of the given matrix is:

$\begin{array}{l} A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] \\ A^{- 1} = \frac{1}{(4) (- 1) - (- 5) (2)} [\begin{matrix} - 1 & - (- 5) \\ - (2) & 4 \end{matrix}] \\ A^{- 1} = \frac{1}{- 4 + 10} [\begin{matrix} - 1 & 5 \\ - 2 & 4 \end{matrix}] \\ A^{- 1} = \frac{1}{6} [\begin{matrix} - 1 & 5 \\ - 2 & 4 \end{matrix}] \\ A^{- 1} = [\begin{matrix} - \frac{1}{6} & \frac{5}{6} \\ - \frac{1}{3} & \frac{2}{3} \end{matrix}] \end{array}$

Step 3. Conclusion.

Hence, the inverse of the given matrix is $[\begin{matrix} - \frac{1}{6} & \frac{5}{6} \\ - \frac{1}{3} & \frac{2}{3} \end{matrix}]$ .

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Find the inverse of each matrix, if it exists.
$[\begin{matrix} 4 & - 5 \\ 2 & - 1 \end{matrix}]$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Conclusion.

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Find the inverse of each matrix, if it exists.4−52−1

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find the inverse of each matrix, if it exists.
$[\begin{matrix} 4 & - 5 \\ 2 & - 1 \end{matrix}]$