Q. 37

# 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 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 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 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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