Q. 37

# Prove that: (i) adj I = I (ii) adj O = O (iii) I^{-1} = I.

Answer :

(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:

a_{11} (co – factor of 1) = (-1)^{1+1}(1) = (-1)^{2}(1) = 1

a_{12} (co – factor of 0) = (-1)^{1+2}(0) = (-1)^{3}(0) = 0

a_{21} (co – factor of 0) = (-1)^{2+1}(0) = (-1)^{3}(0) = 0

a_{22} (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:

a_{11} (co – factor of 0) = (-1)^{1+1}(0) = 0

a_{12} (co – factor of 0) = (-1)^{1+2}(0) = 0

a_{21} (co – factor of 0) = (-1)^{2+1}(0) = 0

a_{22} (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

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