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:


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


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