Question

Let A be a 2 × 2 matrix with elements \(a_{11} = a_{12} = a_{21} = +1\) and \(a_{22} = −1\). Then the eigenvalues of matrix A² are

• A. 1024 and -1024
• B. 512√2 and −512√2
• C. 1024√2 and −1024√2
• D. 4√2 and −4√2

Hint:

For powers of matrices, calculate eigenvalues of the original matrix first and raise them to the required power.

Answer 1

The Correct Option is B

The eigenvalues for matrix A are obtained from the characteristic equation: det(A − λI) = 0. Using the property that A^n's eigenvalues are the n-th power of A's eigenvalues, the eigenvalues for A^19 were computed. The resulting eigenvalues are 512√2 and −512√2.