Chapter 11: Q. 12 (page 795)

find the inverse of each nonsingular matrix.
$[\begin{matrix} 1 & - 1 & 1 \\ 2 & 5 & - 1 \\ 2 & 3 & 0 \end{matrix}]$

Short Answer

Expert verified

find the inverse of each nonsingular matrix.

$[\begin{matrix} 3 & 3 & - 4 \\ - 2 & - 2 & 3 \\ - 4 & - 5 & 7 \end{matrix}]$

Step by step solution

Step 1. Given

$B = [\begin{matrix} 1 & - 1 & 1 \\ 2 & 5 & - 1 \\ 2 & 3 & 0 \end{matrix}]$

Step 2. Calculation of Deterrminant

$|\begin{matrix} 1 & - 1 & 1 \\ 2 & 5 & - 1 \\ 2 & 3 & 0 \end{matrix}| = 1 (5 \times 0 - (- 1) \times 3) - (- 1) (2 \times 0 - (- 1) \times 2) + (2 \times 3 - 5 \times 2) = 1 (3) + 1 (2) + 1 (- 4) = 3 - 2 - 4 = - 3 t h u s m a t r i x i s n o n - \sin g u l a r$

Step 3. Calculation for Augmented matrix

$[\begin{matrix} 1 & - 1 & 1 & 1 & 0 & 0 \\ 2 & 5 & - 1 & 0 & 1 & 0 \\ 2 & 3 & 0 & 0 & 0 & 1 \end{matrix}] A p p l y i n g r o w t r a n s f o r m a t i o n [\begin{matrix} 1 & - 1 & 1 & 1 & 0 & 0 \\ 2 & 5 & - 1 & 0 & 1 & 0 \\ 2 & 3 & 0 & 0 & 0 & 1 \end{matrix}] \overset{R_{2} - 2 R_{1}}{\to [\begin{matrix} 1 & - 1 & 1 & 1 & 0 & 0 \\ 2 & 5 & - 1 & 0 & 1 & 0 \\ 0 & 5 & - 2 & - 2 & 0 & 1 \end{matrix}]} \overset{\frac{R_{2}}{7}}{\to} [\begin{matrix} 1 & - 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & - \frac{3}{7} & - \frac{2}{7} & \frac{1}{7} & 0 \\ 0 & 5 & - 2 & - 2 & 0 & 1 \end{matrix}] \overset{R_{1} + R_{2}}{\to} [\begin{matrix} 1 & 0 & \frac{4}{7} & \frac{5}{7} & \frac{1}{7} & 0 \\ 0 & 1 & - \frac{3}{7} & - \frac{2}{7} & \frac{1}{7} & 0 \\ 0 & 5 & - 2 & - 2 & 0 & 1 \end{matrix}] \overset{R_{3} - 5 R_{2}}{\to} [\begin{matrix} 1 & 0 & \frac{4}{7} & \frac{5}{7} & \frac{1}{7} & 0 \\ 0 & 1 & - \frac{3}{7} & - \frac{2}{7} & \frac{1}{7} & 0 \\ 0 & 0 & \frac{1}{7} & - \frac{4}{7} & - \frac{5}{7} & 1 \end{matrix}] \overset{7 R_{3}}{\to} [\begin{matrix} 1 & 0 & \frac{4}{7} & \frac{5}{7} & \frac{1}{7} & 0 \\ 0 & 1 & - \frac{3}{7} & - \frac{2}{7} & \frac{1}{7} & 0 \\ 0 & 0 & 1 & - 4 & - 5 & 7 \end{matrix}] \overset{R_{1} - \frac{4}{7} R_{3}}{\to} [\begin{matrix} 1 - \frac{4}{7} (0) & 0 - \frac{4}{7} (0) & \frac{4}{7} - \frac{4}{7} (1) & \frac{5}{7} - \frac{4}{7} (- 4) & \frac{1}{7} - \frac{4}{7} (- 5) & 0 - \frac{4}{7} (7) \\ 0 & 1 & - \frac{3}{7} & - \frac{2}{7} & \frac{1}{7} & 0 \\ 0 & 0 & 1 & - 4 & - 5 & 7 \end{matrix}] \overset{R_{2} + \frac{3}{7} R_{3}}{\to} [\begin{matrix} 1 & 0 & 0 & 3 & 3 & - 4 \\ 0 & 1 & 0 & - 2 & - 2 & 0 \\ 0 & 0 & 1 & - 4 & - 5 & 7 \end{matrix}] T h e r e f o r e t h e i n v e r s e i s [\begin{matrix} 3 & 3 & - 4 \\ - 2 & - 2 & 0 \\ - 4 & - 5 & 7 \end{matrix}]$

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find the inverse of each nonsingular matrix.
$[\begin{matrix} 1 & - 1 & 1 \\ 2 & 5 & - 1 \\ 2 & 3 & 0 \end{matrix}]$

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation of Deterrminant

Step 3. Calculation for Augmented matrix

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find the inverse of each nonsingular matrix.1-1125-1230

Short Answer

Step by step solution

Step 1. Given

Step 2. Calculation of Deterrminant

Step 3. Calculation for Augmented matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Decision Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

find the inverse of each nonsingular matrix.
$[\begin{matrix} 1 & - 1 & 1 \\ 2 & 5 & - 1 \\ 2 & 3 & 0 \end{matrix}]$