Certainly! Let's go through each part step-by-step to arrive at the solution.
### Part a) Finding [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
We are given the matrix equation:
[tex]\[
\left( \begin{array}{cc} 3x + y & 0 \\ 0 & x - y \end{array} \right) - \left( \begin{array}{cc} 7 & 0 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)
\][/tex]
To solve this, we equate the corresponding elements from the left-hand side and the right-hand side of the equation. This gives us the following equations:
[tex]\[
3x + y - 7 = 0
\][/tex]
[tex]\[
x - y - 1 = 0
\][/tex]
From the first equation, we have:
[tex]\[
3x + y = 7
\][/tex]
From the second equation, we have:
[tex]\[
x - y = 1
\][/tex]
Now we solve these two equations simultaneously. Adding these two equations, we get:
[tex]\[
3x + y + x - y = 7 + 1 \implies 4x = 8 \implies x = 2
\][/tex]
Substitute [tex]\( x = 2 \)[/tex] back into the second equation:
[tex]\[
2 - y = 1 \implies y = 1
\][/tex]
Therefore, the solutions are [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
### Part b) Evaluating the Matrix Expression
We need to evaluate the following expression:
[tex]\[
\left( \begin{array}{cc} 3 & 5 \\ 2 & 7 \end{array} \right) + 2 \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) - \left( \begin{array}{cc} 5 & 9 \\ 8 & 15 \end{array} \right)
\][/tex]
First, compute the scalar multiplication:
[tex]\[
2 \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) = \left( \begin{array}{cc} 2 \times 1 & 2 \times 2 \\ 2 \times 3 & 2 \times 4 \end{array} \right) = \left( \begin{array}{cc} 2 & 4 \\ 6 & 8 \end{array} \right)
\][/tex]
Next, perform the matrix addition and subtraction:
[tex]\[
\left( \begin{array}{cc} 3 & 5 \\ 2 & 7 \end{array} \right) + \left( \begin{array}{cc} 2 & 4 \\ 6 & 8 \end{array} \right) = \left( \begin{array}{cc} 3 + 2 & 5 + 4 \\ 2 + 6 & 7 + 8 \end{array} \right) = \left( \begin{array}{cc} 5 & 9 \\ 8 & 15 \end{array} \right)
\][/tex]
Subtracting the final matrix:
[tex]\[
\left( \begin{array}{cc} 5 & 9 \\ 8 & 15 \end{array} \right) - \left( \begin{array}{cc} 5 & 9 \\ 8 & 15 \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)
\][/tex]
Therefore, the result of the matrix expression is:
[tex]\[
\left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)
\][/tex]
### Part c) Evaluating [tex]\( 0^2 + 30 \)[/tex]
It is given to find [tex]\( 0^2 + 30 \)[/tex].
Let's compute that directly:
[tex]\[
0^2 = 0
\][/tex]
Adding 30:
[tex]\[
0 + 30 = 30
\][/tex]
Therefore, the result is:
[tex]\[
30
\][/tex]
### Summary
- The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
- The result of the matrix evaluation is [tex]\( \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) \)[/tex].
- The result of [tex]\( 0^2 + 30 \)[/tex] is [tex]\( 30 \)[/tex].
