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Prove that: (i) \operatorname{adj} I=I (ii) \operatorname{adj} O=O (iii) I^{-1}=I.
Adjoint and Inverse of a Matrix | CBSE | Class 12 | Exercise 7A | Maths | RS Aggarwal
Prove that: (i) \operatorname{adj} I=I (ii) \operatorname{adj} O=O (iii) I^{-1}=I.

Solution:

(i) We need to compute a d j I :
We have I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)
Find: the adjoint of the given matrix.

Step: 1 Find the minor matrix of I.
M_{I}=\left(\begin{array}{ll} |1| & |0| \\ |0| & |1| \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)

Step: 2 Find the co-factor matrix of I.
C_{I}=\left(\begin{array}{ll} (-1)^{1+1}(1) & (-1)^{1+2}(0) \\ (-1)^{2+1}(0) & (-1)^{2+2}(1) \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)

Step: 3 By transpose of C_{I} we will have \operatorname{adj} I.
\operatorname{adj} I=C_{I}^{T}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)^{T}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)=I

(ii) We need to compute adj O :
We have O=\left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right)
Find: the adjoint of the given matrix.

Step: 1 Find the minor matrix of O.
M_{O}=\left(\begin{array}{ll} |0| & |0| \\ |0| & |0| \end{array}\right)=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)

Step: 2 Find the co-factor matrix of O.
C_{O}=\left(\begin{array}{ll} (-1)^{1+1}(0) & (-1)^{1+2}(0) \\ (-1)^{2+1}(0) & (-1)^{2+2}(0) \end{array}\right)=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)

Step: 3 By transpose of C_{O} we will have \operatorname{adj} O.
\operatorname{adj} O=C_{O}^{T}=\left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right)^{T}=\left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right)=0

(iii) We need to compute I^{-1} :
We have |I|=\left|\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right|=1 \neq 0 so inverse of given matrix exist.
The inverse of given matrix is
I^{-1}=\frac{1}{|I|} \text { adj } I=\frac{1}{1}\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)=I

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