InterviewSolution
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InterviewSolution
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Match the following lists : |
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Answer» `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` |
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