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 matrix that is a multiple of the identity matrix. The general form of a scalar matrix is \( \lambda I \), where \( I \) is the identity matrix and \( \lambda \) is a scalar. In this case, the given matrix is: \[ \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}, \] which can be written as: \[ 7 \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = 7I. \] This shows that the matrix is indeed a scalar matrix with \( \lambda = 7 \). Now, the expression \( x^y \) is essentially referring to a scalar matrix raised to a power. Since scalar matrices are multiples of the identity matrix, when raised to any power, the result will be another scalar matrix with the scalar raised to that power on the diagonal. For example: \[ (7I)^n = 7^n \times I. \] This means that if we consider \( x^y \) to be a scalar matrix, it is equal to 1, because the diagonal elements are equal and the scalar matrix raised to any power remains the same. Therefore, the correct answer is: \[ \boxed{1}. \]