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Find the inverse of each of the following matrices.
\begin{aligned} &(i)\left[\begin{array}{ccc} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{array}\right] \\ \end{aligned}
\begin{aligned} &(ii)\left[\begin{array}{ccc} 0 & 0 & -1 \\ 3 & 4 & 5 \\ -2 & -4 & -7 \end{array}\right] \end{aligned}
Adjoint and Inverse of a Matrix | CBSE | Class 12 | Exercise 7.1 | Maths | RD Sharma
Find the inverse of each of the following matrices.
\begin{aligned} &(i)\left[\begin{array}{ccc} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{array}\right] \\ \end{aligned}
\begin{aligned} &(ii)\left[\begin{array}{ccc} 0 & 0 & -1 \\ 3 & 4 & 5 \\ -2 & -4 & -7 \end{array}\right] \end{aligned}

Solution:

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

|\mathrm{A}|=0\left|\begin{array}{ll}-3 & 0 \\ -3 & 4\end{array}\right|-1\left|\begin{array}{ll}4 & 4 \\ 3 & 4\end{array}\right|-1\left|\begin{array}{ll}4 & -3 \\ 3 & -3\end{array}\right|

=0-1(16-12)-1(-12+9)

=-4+3

=-1 \neq 0

Thus, A^{-1} exists

Cofactors of A are

C_{11}=0

C_{21}=-1

C_{31}=1

C_{12}=-4

C_{22}=3

C_{32}=-4

C_{13}=-3

C_{23}=3

C_{33}=-4

It is known that adj A=\left[\begin{array}{lll}C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}\end{array}\right]^{T}

=\left[\begin{array}{ccc}0 & -4 & -3 \\ -1 & 3 & 3 \\ 1 & -4 & -4\end{array}\right]^{\mathrm{T}}

Therefore, adj A=\left[\begin{array}{ccc}0 & -1 & 1 \\ -4 & 3 & -4 \\ -3 & 3 & -4\end{array}\right]

Now, A^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj} A

Therefore, A^{-1}=\frac{1}{-1}\left[\begin{array}{ccc}0 & -1 & 1 \\ -4 & 3 & -4 \\ -3 & 3 & -4\end{array}\right]

Thus, \mathrm{A}^{-1}=\left[\begin{array}{ccc}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right]

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

|\mathrm{A}|=0\left|\begin{array}{cc}4 & 5 \\ -4 & -7\end{array}\right|-0\left|\begin{array}{cc}3 & 5 \\ -2 & -7\end{array}\right|-1\left|\begin{array}{cc}3 & 4 \\ -2 & -4\end{array}\right|

=0-0-1(-12+8)

=4 \neq 0

Thus, A^{-1} exists

Cofactors of A are

C_{11}=-8

\mathrm{C}_{21}=4

C_{31}=4

\mathrm{C}_{12}=11

C_{22}=-2

C_{32}=-3

\mathrm{C}_{13}=-4

C_{23}=0

\mathrm{C}_{33}=0

It is known that adj A=\left[\begin{array}{lll}C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}\end{array}\right]^{\mathrm{T}}

=\left[\begin{array}{ccc}8 & 11 & -4 \\ 4 & -2 & 0 \\ 4 & -3 & 0\end{array}\right]^{\mathrm{T}}

Therefore, \operatorname{adj} \mathrm{A}=\left[\begin{array}{ccc}8 & 4 & 4 \\ 11 & -2 & -3 \\ -4 & 0 & 0\end{array}\right]

Now, A^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj} A

Therefore, A^{-1}=\frac{1}{4}\left[\begin{array}{ccc}8 & 4 & 4 \\ 11 & -2 & -3 \\ -4 & 0 & 0\end{array}\right]

Thus, A^{-1}=\left[\begin{array}{ccc}2 & 1 & 1 \\ \frac{11}{4} & \frac{-1}{2} & \frac{-3}{4} \\ -1 & 0 & 0\end{array}\right]

i

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