Question

Which of the following relation is correct?

(a) A = A^T

(b) A = -A^T

(c) A = A^2T

(d) A = A^T/2

I had been asked this question by my school principal while I was bunking the class.

Question is taken from Diagonalization Powers of a Matrix in section Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

Correct choice is (a) A = A^T

For explanation I would say: To prove that A = A^T, let us consider an example,

\(A = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix}\)

⎸A – λI ⎸ = 0

\( \begin{vmatrix} 1-λ & 0 \\ 2 & 3-λ \end{vmatrix} = 0\)

(1-λ) (3-λ) = 0

3 – λ – 3λ +λ^2 = 0

λ^2 – 4λ + 3 = 0

(λ – 3) (λ – 1) = 0

λ = 1, 3

Consider \(A^T = \begin{bmatrix}1 & 2\\ 0 & 3 \end{bmatrix}\)

⎸A – λI ⎸ = 0

\(\begin{bmatrix}1-λ & 2 \\ 0 & 3-λ \end{bmatrix} = 0\)

(1-λ) (3-λ) = 0, which is similar to the result obtained for A, hence the eigen values are same.