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Chapter 10: Q85. (page 530)

Find the inverse of the matrix
$[\begin{matrix} - 5 & 6 \\ - 11 & 3 \end{matrix}]$
if it exists.

Short Answer

Expert verified

The inverse of the given matrix is $\frac{1}{|51|} [\begin{matrix} 3 & 鈭� 6 \\ 11 & 鈭� 5 \end{matrix}]$

Step by step solution

01

Step 1. Write down the given equation.

The given matrix is say

P = [\begin{matrix} - 5 & 6 \\ - 11 & 3 \end{matrix}]

.

02

Step 2. Concept used.

The inverse of the matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

exits if $|A| 鈮� 0$ . And inverse of $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

is given by the formula:

$A^{鈭� 1} = \frac{1}{|A|} [\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix}]$ where $|A| 鈮� 0$

03

Step 3. Evaluating inverse of given matrix.

The inverse of the matrix is evaluated as:

$\begin{array}{l} P = [\begin{matrix} 鈭� 5 & 6 \\ 鈭� 11 & 3 \end{matrix}] .... (Given) \\ |P| = 鈭� 15 + 66 \\ = 51 \end{array}$

Now evaluating inverse of

P = [\begin{matrix} 鈭� 5 & 6 \\ 鈭� 11 & 3 \end{matrix}]

by the formula stated above.

$P^{鈭� 1} = \frac{1}{|51|} [\begin{matrix} 3 & 鈭� 6 \\ 11 & 鈭� 5 \end{matrix}]$

04

Step 4. Conclusion.

Hence the inverse of the given matrix is

$\frac{1}{|51|} [\begin{matrix} 3 & 鈭� 6 \\ 11 & 鈭� 5 \end{matrix}]$

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