Match the following lists :

1.	Match the following lists :
Answer» `a rarr r, b rarr s, c rarr p, r, d rarr p, q, r, s`. a. Since A is idempotent, `A^(2)=A^(3)=A^(4)= =A`. Now, `(A+I)^(n)=I+ .^(n)C_(1)A+ .^(n)C_(2)A^(2)+ +.^(n)C_(n)A^(n)` `=I+.^(n)C_(1)A+.^(n)C_(2)A+ + .^(n)C_(n)A` `=I+(.^(n)C_(1)+.^(n)C_(1)+ +.^(n)C_(n))A` `=I+(2^(n)-1)A` `implies 2^(n)-1=127` `implies n=7` b. We have, `(I-A) (I+A+A^(2)+ + A^(7))` `=I+A+A^(2)+ +A^(7)+ (-A-A^(2)-A^(3)-A^(4)-A^(8))` `=I-A^(8)` `=I ("if "A^(8)=O)` c. Here matrix A is skew-symmetric and since `\|A\|=\|-A^(T)\|=(-1)^(n)\|A\|`, so `\|A\| (1-(-1)^(n))=0`. As n is odd, hence `\|A\|=0`, Hence, A is singular. d. If A is symmetric, `A^(-1)` is also symmetric for matrix of any order.

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