Match the following lists :

Explanation

`a rarr s, b rarr p, c rarr q, d rarr r`.
a. Since A is idempotent, we have
`A^(2)=A`
`implies A^(3)=A A^(2)=A A=A^(2)=A, A^(4)=A A^(3)=A A=A^(2)=A`
`implies A^(n)=A`
`implies (I-A)^(n)=^(n)C_(0)I-.^(n)C_(1)A+.^(n)C_(2)A^(2)- .^(n)C_(3)A^(3)+`
`= I+(-^(n)C_(1)+ .^(n)C_(2)-^(n)C_(3)+)A`
`=I+[(.^(n)C_(0)-^(n)C_(1)+^(n)C_(3)+)-^(n)C_(0)]A`
`=I-A`
b. A is involuntary. Hence,
`A^(2)=I`
`implies A^(3)=A^(5)= = A` and `A^(2)=A^(4)=A^(6)= =I`
`implies (I-A)^(n)=^(n)C_(0)I-^(n)C_(1)A+ .^(n)C_(2)A^(2)-^(n)C_(3)A^(3)+`
`=^(n)C_(0)I-^(n)C_(1)A+.^(n)C_(1)I-.^(n)C_(3)A+^(n)C_(4)I-`
`=(.^(n)C_(0)+.^(n)C_(2)+.^(n)C_(4)+)I-(.^(n)C_(1)A+.^(n)C_(3)+.^(n)C_(5)+)A`
`=2^(n-1) (I-A)`
`implies [(I-A)^(n)]A^(-1)=2^(n-1) (I-a)A^(-1)=2^(n-1) (A^(-1)-I)`
c. If A is nilpotent of index 2, then
`A^(2)=A^(3)=A^(4)=A^(n)=O`
`implies (I-A)^(n)=^(n)C_(0)I- .^(n)C_(1)A+.^(n)C_(2)A^(2)- .^(n)C_(3)A^(3)+`
`=I-nA+O+O+`
`=I-nA`
d. A is orthogonal. Hence,
`A A^(T)=I`
`implies (A^(T))^(-1) =A`