Question

If \( \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix} \) is a scalar matrix, then \( x^y \) is equal to:

• A. 0
• B. 1
• C. 7
• D. \( \pm 7 \)

Hint:

For scalar matrices, the elements on the diagonal are equal, and raising them to any power will result in a matrix with those powers on the diagonal.

Answer 1

The Correct Option is B

A scalar matrix is a multiple of the identity matrix, denoted as \( \lambda I \), where \( I \) is the identity matrix and \( \lambda \) is a scalar. The provided matrix \[\begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}\] can be expressed as \( 7 \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = 7I \). Thus, it is a scalar matrix with \( \lambda = 7 \). The expression \( x^y \) represents a scalar matrix raised to a power. When a scalar matrix \( \lambda I \) is raised to any power \( n \), the result is \( (\lambda I)^n = \lambda^n I \), which is also a scalar matrix. If \( x^y \) is to be interpreted as a scalar matrix, and given that the scalar matrix raised to any power remains constant, it implies that \( x^y \) equals 1. Therefore, the result is:\[\boxed{1}.\]