Question

Prove that: 

(i) adj I = I 

(ii) adj O = O 

(iii) I^-1 = I.

Answer 1

(i) To Prove: adj I = I

We know that, I means the Identity matrix

Let I is a 2 × 2 matrix

I = \(\begin{bmatrix} 1& 0 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\)

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