To determine for what values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] the matrix
[tex]\[ M = \begin{pmatrix} 8 & x - y \\ x + y & 8 \end{pmatrix} \][/tex]
is a scalar matrix, we must understand the properties of a scalar matrix.
A scalar matrix is a diagonal matrix in which all the diagonal elements are equal. In other words:
1. The diagonal elements of the matrix must be equal.
2. The off-diagonal elements must be zero.
Step-by-Step Solution:
1. Verify the Diagonal Elements:
The given matrix is:
[tex]\[
M = \begin{pmatrix} 8 & x - y \\ x + y & 8 \end{pmatrix}
\][/tex]
In a scalar matrix, the diagonal elements are already equal as given ([tex]\( M_{11} = M_{22} = 8 \)[/tex]), so this condition is satisfied.
2. Ensure the Off-Diagonal Elements are Zero:
The off-diagonal elements must be zero. Hence, we set up the following equations:
[tex]\[
x - y = 0
\][/tex]
[tex]\[
x + y = 0
\][/tex]
3. Solve the First Off-Diagonal Equation:
From the first equation:
[tex]\[
x - y = 0
\][/tex]
Solving for [tex]\( y \)[/tex], we get:
[tex]\[
y = x
\][/tex]
4. Solve the Second Off-Diagonal Equation:
Substitute [tex]\( y = x \)[/tex] into the second equation:
[tex]\[
x + y = 0
\][/tex]
Substituting [tex]\( y = x \)[/tex], we get:
[tex]\[
x + x = 0
\][/tex]
[tex]\[
2x = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 0
\][/tex]
5. Find the Corresponding Value of [tex]\( y \)[/tex]:
Since [tex]\( y = x \)[/tex], substituting [tex]\( x = 0 \)[/tex] gives:
[tex]\[
y = 0
\][/tex]
Therefore, for the matrix [tex]\( \begin{pmatrix} 8 & x - y \\ x + y & 8 \end{pmatrix} \)[/tex] to be a scalar matrix, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] must be [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex].
