Let A be a `nxxn` matrix such that`A ^(n) = alpha A,` where `alpha ` is a
real number different from 1 and - 1. The matrix `A + I_(n)` is

Let A be an nxxn matrix such that A+A^(')=O_(nxxn) then |I+lamdaA^(2)|geM where 3M+1 is __

The number of nxxn matrix A and B such that AB - BA = I is. . .

(aA)^(-1)=1/aA^(-1) where a is any real number and A is a square matrix.

Let A be a symmetric matrix, then A^(m), m in N is a

Let A be a non - singular square matrix such that A^(2)=A satisfying (I-0.8A)^(-1)=I-alphaA where I is a unit matrix of the same order as that of A, then the value of -4alpha is equal to

The value of alpha^(-n)+alpha^(-2n), n in N and alpha is a non-real cube root of unity, is

If f(x) is a polynomial of degree n(gt2) and f(x)=f(alpha-x) , (where alpha is a fixed real number ), then the degree of f'(x) is

Let A be a square matrix of order n . l = maximum number of different entries if A is a upper triangular matrix. m = minimum number of zeros if A is a triangular matrix. p = minimum number of zeros if A is a diagonal matrix. If l+2m=2p+1 , then n is :

Let A = [a_(ij)] " be a " 3 xx3 matrix and let A_(1) denote the matrix of the cofactors of elements of matrix A and A_(2) be the matrix of cofactors of elements of matrix A_(1) and so on. If A_(n) denote the matrix of cofactros of elements of matrix A_(n -1) , then |A_(n)| equals

If A is an idempotent matrix satisfying (I-0.4A)^(-1)=I-alphaA where I is the unit matrix of the same order as that of A then the value of alpha is