Chapter 11: Q. 36 (page 756)

In Problems $31 - 40$ , each matrix is nonsingular. Find the inverse of each matrix.
$[\begin{array}{l} b & 3 \\ b & 2 \end{array}] b \neq 0$

Short Answer

Expert verified

The inverse of the matrix $[\begin{array}{l} b & 3 \\ b & 2 \end{array}] b \neq 0$ is, $[\begin{matrix} \frac{- 2}{b} & \frac{3}{b} \\ 1 & - 1 \end{matrix}]$ .

Step by step solution

Step 1. Given information

$[\begin{array}{l} b & 3 \\ b & 2 \end{array}] b \neq 0$

We have to find the inverse of the given matrix.

Step 2. First find the augmented matrix A|I3.

$[A | I_{3}] \to [\begin{array}{l} b & 3 & 1 & 0 \\ b & 2 & 0 & 1 \end{array}]$

Now transform it into $[I_{3} | A^{- 1}]$ using the row transformations.

$[\begin{array}{l} b & 3 & 1 & 0 \\ b & 2 & 0 & 1 \end{array}] \to \{R_{2} \to R_{2} - R_{1}\} [\begin{matrix} b & 3 & 1 & 0 \\ 0 & - 1 & - 1 & 1 \end{matrix}] \to \{R_{1} \to \frac{R_{1}}{b}\} [\begin{matrix} 1 & \frac{3}{b} & \frac{1}{b} & 0 \\ 0 & - 1 & - 1 & 1 \end{matrix}] \to \{R_{2} \to - R_{2}\} [\begin{matrix} 1 & \frac{3}{b} & \frac{1}{b} & 0 \\ 0 & 1 & 1 & - 1 \end{matrix}] \to \{R_{1} \to R_{1} - \frac{3}{b} R_{2}\} [\begin{matrix} 1 & 0 & - \frac{2}{b} & \frac{3}{b} \\ 0 & 1 & 1 & - 1 \end{matrix}]$

Step 3. The augmented matrix A|In is now in the reduced row echelon form and the identity matrix src="data:image/svg+xml;base64,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" role="math" localid="1647448841999" I2 is on the left side of the vertical bar and A-1 is on the right side of the vertical bar.

$A^{- 1} = [\begin{matrix} - \frac{2}{b} & \frac{3}{b} \\ 1 & - 1 \end{matrix}]$ .

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In Problems $31 - 40$ , each matrix is nonsingular. Find the inverse of each matrix.
$[\begin{array}{l} b & 3 \\ b & 2 \end{array}] b \neq 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. First find the augmented matrix A|I3.

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In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.b3b2b≠0

Short Answer

Step by step solution

Step 1. Given information

Step 2. First find the augmented matrix A|I3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems $31 - 40$ , each matrix is nonsingular. Find the inverse of each matrix.
$[\begin{array}{l} b & 3 \\ b & 2 \end{array}] b \neq 0$