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  • Provide Solution for RD Sharma Class 12 Chapter 6 Adjoint and Inverse Matrix Exercise 6.1 Question 12

Provide Solution for RD Sharma Class 12 Chapter 6 Adjoint and Inverse Matrix Exercise 6.1 Question 12

Answers (1)

Answer:

Hence proved  2A^{-1}=9 I-A

Hint:

Here, we use basic concept of determinant and inverse of matrix

A^{-1}=\frac{1}{|A|} \times \operatorname{Adj}(A)

Given:

A=\left[\begin{array}{cc} 2 & -3 \\ -4 & 7 \end{array}\right]

Solution:

Here let’s find   |A|, A d j(A) \& A^{-1}

\begin{aligned} &|A|=\left|\begin{array}{cc} 2 & -3 \\ -4 & 7 \end{array}\right|=14-12=2 \\ &\operatorname{Adj}(A)=\left[\begin{array}{cc} 7 & 3 \\ 4 & 2 \end{array}\right] \\ &A^{-1}=\frac{1}{2}\left[\begin{array}{ll} 7 & 3 \\ 4 & 2 \end{array}\right] \end{aligned}

To show  2A^{-1}=9 I-A

2 A^{-1}=2 \times \frac{1}{2}\left[\begin{array}{ll} 7 & 3 \\ 4 & 2 \end{array}\right]=\left[\begin{array}{ll} 7 & 3 \\ 4 & 2 \end{array}\right]                          (1)

9 I-A=\left[\begin{array}{ll} 9 & 0 \\ 0 & 9 \end{array}\right]-\left[\begin{array}{cc} 2 & -3 \\ -4 & 7 \end{array}\right]=\left[\begin{array}{ll} 7 & 3 \\ 4 & 2 \end{array}\right]                 (2)

From equation (1) and (2)

Hence,  2A^{-1}=9 I-A

 

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