For what values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is the matrix [tex]\(\left(\begin{array}{cc}8 & x-y \\ x+y & 8\end{array}\right)\)[/tex] a scalar matrix?



Answer :

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].