Match the following lists :

Explanation

`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.