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Find the inverse of each of the following matrices:
(i) \left[\begin{array}{cc}a & b \\ c & \frac{1+b c}{a_{7}}\end{array}\right]
(ii)\left[\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right]
Adjoint and Inverse of a Matrix | CBSE | Class 12 | Exercise 7.1 | Maths | RD Sharma
Find the inverse of each of the following matrices:
(i) \left[\begin{array}{cc}a & b \\ c & \frac{1+b c}{a_{7}}\end{array}\right]
(ii)\left[\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right]

Solution:

(i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero.

\left.\begin{array}{l} \text { Now, }|\mathrm{A}|=\frac{\mathrm{a}+\mathrm{abc}}{\mathrm{a}}-\mathrm{bc}=\frac{\mathrm{a}+\mathrm{abc}-\mathrm{abc}}{\mathrm{a}}=1 \neq 0 \\ \text { Therefore, } \mathrm{A}^{-1} \text { exists. } \\ \text { Cofactors of } \mathrm{A} \text { are } \\ C_{11}=\frac{1+\mathrm{bc}}{\mathrm{a}} \\ C_{12}=-\mathrm{C} \\ C_{21}=-\mathrm{b} \\ C_{22}=\mathrm{a} \\ \text { As, } \operatorname{adj} \mathrm{A}=\left[\begin{array}{ll} \mathrm{C}_{11} & \mathrm{C}_{12} \\ \mathrm{C}_{21} & \mathrm{C}_{22} \end{array}\right]^{\mathrm{T}} \\ \mathrm{A}^{\mathrm{T}} \end{array}\right

(ii) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero.

Now, |\mathrm{A}|=2+15=17 \neq 0

Therefore, A^{-1} exists.

Cofactors of A are

C_{11}=1

C_{12}=3

C_{21}=-5

C_{22}=2

\text { As, } \operatorname{adj} A=\left[\begin{array}{ll} C_{11} & C_{12} \\ C_{21} & C_{22} \end{array}\right]^{T}

(\operatorname{adj} A)=\left[\begin{array}{cc} 1 & 3 \\ -5 & 2 \end{array}\right]^{T}

=\left[\begin{array}{cc} 1 & -5 \\ 3 & 2 \end{array}\right]

\text { Now, } A^{-1}=\frac{1}{|A|} \cdot \operatorname{adj} A

A^{-1}=\frac{1}{17}\left[\begin{array}{cc} 1 & -5 \\ 3 & 2 \end{array}\right]

i

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