Chapter 11: Q. 11 (page 795)

find the inverse of each nonsingular matrix.
$[\begin{matrix} 3 & 2 \\ 5 & 4 \end{matrix}]$

Short Answer

Expert verified

The inverse of the given matrix is

$[\begin{matrix} 2 & - 1 \\ - \frac{5}{2} & \frac{3}{2} \end{matrix}]$

Step by step solution

Step 1. Given

$A = [\begin{matrix} 3 & 2 \\ 5 & 4 \end{matrix}]$

Step 2. Calculation of A|I2

Given A as $2 \times 2$ matrix therefore $[A | I_{2}]$ where $I_{2}$ is identity matrix of second order.

$[A | I_{2}] = [\begin{matrix} 3 & 2 & 1 & 0 \\ 5 & 4 & 0 & 1 \end{matrix}]$

Step 3. Calculation for transformation

Transforming matrix $[A | I_{2}]$

$p e r f o r m t h e r o w o p e r a t i o n R_{1} = \frac{1}{3} r_{1} & R_{2} = r_{2} - 5 r_{1} [\begin{matrix} 3 & 2 & 1 & 0 \\ 5 & 4 & 0 & 1 \end{matrix}] \to [\begin{matrix} 1 & \frac{2}{3} & \frac{1}{3} & 0 \\ 5 & 4 & 0 & 1 \end{matrix}] \to [\begin{matrix} 1 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{2}{3} & - \frac{5}{3} & 1 \end{matrix}]$

Step 4. Final Calculation

Perform

$R_{1} = r_{1} - r_{2} & R_{2} = \frac{3}{2} r_{2} [\begin{matrix} 1 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{2}{3} & - \frac{5}{3} & 1 \end{matrix}] \to [\begin{matrix} 1 & 0 & 2 & - 1 \\ 0 & \frac{2}{3} & - \frac{5}{3} & 1 \end{matrix}] \to [\begin{matrix} 1 & 0 & 2 & - 1 \\ 0 & 1 & - \frac{5}{2} & \frac{3}{2} \end{matrix}]$

$T h u s w e s e e t h e i n v e r s e a s [\begin{matrix} 2 & - 1 \\ - \frac{5}{2} & \frac{3}{2} \end{matrix}]$

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find the inverse of each nonsingular matrix.
$[\begin{matrix} 3 & 2 \\ 5 & 4 \end{matrix}]$

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation of A|I2

Step 3. Calculation for transformation

Step 4. Final Calculation

One App. One Place for Learning.

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find the inverse of each nonsingular matrix.3254

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation of A|I2

Step 3. Calculation for transformation

Step 4. Final Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

find the inverse of each nonsingular matrix.
$[\begin{matrix} 3 & 2 \\ 5 & 4 \end{matrix}]$