GATE CE 2005 linear algebra

Question

Consider the system of linear equations A(n*n)X(n*1) = λ(n*1) where λ is a scalar. Let (λi , Xi) be an eigen pair of an eigen value and its corresponding eigen vector for a real matrix A. Let I be a n*n unit matrix. Which one of the following statements is not correct?

(A) for a homogeneous n*n system of linear equations (A-λI)X=0  having a non trivial solution, the rank of (A-λI) is less than n.

(B) for matrix A^m , m being a positive integer, ((λi)^m, (Xi)^m) will be the eigen pair for all i.

(C) if ( A's transpose= A's inverse ) , then mod(λi)=1 for all i.

(D) if (A's transpose = A) , then λi is real for all i

Comments

Whats eigen pair corresponding to eigen vector?

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it is eigen vector corresponding to that eigen value

Answer 1

ANSWER B

Eigen value has some properties.

If $\lambda$ be an Eigen value of a matrix A and X be its corresponding Eigen vector then for Aⁿ the eigen value is $\lambda^{n}$ but the Eigen vector remains same i.e X.

In option A ,non trivial means infinite.A homogeneous system has infinite solutions when rank of coefficient matrix is less than n,where n is the number of unknown variables.Hence it is correct

In option C, A is an orthogonal matrix. Eigen value of orthogonal matrix has unit modulus.

In option D,A is a  symmetric matrix and the eigen values of a real symmetric matrix is always real.

Comments

how does non-trivial mean infinite??