Prove that: (i) adj I = I (ii) adj O = O (iii) I-1 = I.
(i) To Prove: adj I = I
We know that, I means the Identity matrix
Let I is a 2 × 2 matrix
Now, we have to find adj I and for that we have to find co-factors:
a11 (co – factor of 1) = (-1)1+1(1) = (-1)2(1) = 1
a12 (co – factor of 0) = (-1)1+2(0) = (-1)3(0) = 0
a21 (co – factor of 0) = (-1)2+1(0) = (-1)3(0) = 0
a22 (co – factor of 1) = (-1)2+2(1) = (-1)4(1) = 1
Now, adj I = Transpose of co-factor Matrix
Thus, adj I = I
Hence Proved
(ii) To Prove: adj O = O
We know that, O means Zero matrix where all the elements of matrix are 0
Let O is a 2 × 2 matrix
Calculating adj O
Now, we have to find adj O and for that we have to find co-factors:
a11 (co – factor of 0) = (-1)1+1(0) = 0
a12 (co – factor of 0) = (-1)1+2(0) = 0
a21 (co – factor of 0) = (-1)2+1(0) = 0
a22 (co – factor of 0) = (-1)2+2(0) = 0
Now, adj O = Transpose of co-factor Matrix
Thus, adj O = O
Hence Proved
(iii) To Prove: I-1 = I
We know that,
From the part(i), we get adj I
So, we have to find |I|
Calculating |I|
= [1 × 1 – 0]
= 1
Thus, I-1 = I
Hence Proved