To find the given matrix operations, we will systematically go through each required calculation.
1. Matrix Addition ([tex]$A + B$[/tex])
The matrices given are:
[tex]\( A = \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right] \)[/tex]
[tex]\( B = \left[\begin{array}{rr}
-4 & -4 \\
1 & 4
\end{array}\right] \)[/tex]
To add matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we add corresponding elements:
[tex]\[
A + B = \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right] + \left[\begin{array}{rr}
-4 & -4 \\
1 & 4
\end{array}\right] = \left[\begin{array}{cc}
2 + (-4) & 1 + (-4) \\
1 + 1 & 2 + 4
\end{array}\right]
\][/tex]
Simplifying, we get:
[tex]\[
A + B = \left[\begin{array}{cc}
-2 & -3 \\
2 & 6
\end{array}\right]
\][/tex]
2. Matrix Subtraction ([tex]$A - B$[/tex])
To subtract matrix [tex]\(B\)[/tex] from [tex]\(A\)[/tex], we subtract corresponding elements:
[tex]\[
A - B = \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right] - \left[\begin{array}{rr}
-4 & -4 \\
1 & 4
\end{array}\right] = \left[\begin{array}{cc}
2 - (-4) & 1 - (-4) \\
1 - 1 & 2 - 4
\end{array}\right]
\][/tex]
Simplifying, we get:
[tex]\[
A - B = \left[\begin{array}{cc}
6 & 5 \\
0 & -2
\end{array}\right]
\][/tex]
3. Scalar Multiplication ([tex]$4A$[/tex])
To calculate [tex]\(4A\)[/tex], we multiply all elements of matrix [tex]\(A\)[/tex] by 4:
[tex]\[
4A = 4 \times \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right] = \left[\begin{array}{cc}
4 \times 2 & 4 \times 1 \\
4 \times 1 & 4 \times 2
\end{array}\right]
\][/tex]
Simplifying, we get:
[tex]\[
4A = \left[\begin{array}{cc}
8 & 4 \\
4 & 8
\end{array}\right]
\][/tex]
4. Linear Combination ([tex]$4A - 5B$[/tex])
First, we compute [tex]\(5B\)[/tex]:
[tex]\[
5B = 5 \times \left[\begin{array}{rr}
-4 & -4 \\
1 & 4
\end{array}\right] = \left[\begin{array}{rr}
5 \times -4 & 5 \times -4 \\
5 \times 1 & 5 \times 4
\end{array}\right] = \left[\begin{array}{rr}
-20 & -20 \\
5 & 20
\end{array}\right]
\][/tex]
Next, we calculate [tex]\(4A - 5B\)[/tex]:
[tex]\[
4A - 5B = \left[\begin{array}{cc}
8 & 4 \\
4 & 8
\end{array}\right] - \left[\begin{array}{rr}
-20 & -20 \\
5 & 20
\end{array}\right] = \left[\begin{array}{cc}
8 - (-20) & 4 - (-20) \\
4 - 5 & 8 - 20
\end{array}\right]
\][/tex]
Simplifying, we get:
[tex]\[
4A - 5B = \left[\begin{array}{cc}
28 & 24 \\
-1 & -12
\end{array}\right]
\][/tex]
Thus, the results are:
(a) [tex]\(A + B = \left[\begin{array}{cc}
-2 & -3 \\
2 & 6
\end{array}\right]\)[/tex]
(b) [tex]\(A - B = \left[\begin{array}{cc}
6 & 5 \\
0 & -2
\end{array}\right]\)[/tex]
(c) [tex]\(4A = \left[\begin{array}{cc}
8 & 4 \\
4 & 8
\end{array}\right]\)[/tex]
(d) [tex]\(4A - 5B = \left[\begin{array}{cc}
28 & 24 \\
-1 & -12
\end{array}\right]\)[/tex]
