Certainly! Let's tackle the problem step-by-step.
Given the matrices:
[tex]\[ A = \begin{pmatrix}
1 & 4 \\
2 & 3
\end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix}
-4 & -1 \\
-3 & -2
\end{pmatrix} \][/tex]
### Part (a) Find the matrix [tex]\(2A + B\)[/tex]
First, we compute [tex]\( 2A \)[/tex]:
[tex]\[ 2A = 2 \begin{pmatrix}
1 & 4 \\
2 & 3
\end{pmatrix} = \begin{pmatrix}
2 \cdot 1 & 2 \cdot 4 \\
2 \cdot 2 & 2 \cdot 3
\end{pmatrix} = \begin{pmatrix}
2 & 8 \\
4 & 6
\end{pmatrix} \][/tex]
Next, add matrix [tex]\( B \)[/tex] to [tex]\( 2A \)[/tex]:
[tex]\[ 2A + B = \begin{pmatrix}
2 & 8 \\
4 & 6
\end{pmatrix} + \begin{pmatrix}
-4 & -1 \\
-3 & -2
\end{pmatrix} = \begin{pmatrix}
2 + (-4) & 8 + (-1) \\
4 + (-3) & 6 + (-2)
\end{pmatrix} \][/tex]
Performing the addition element-wise:
[tex]\[ 2A + B = \begin{pmatrix}
-2 & 7 \\
1 & 4
\end{pmatrix} \][/tex]
### Part (b) Find a matrix [tex]\(C\)[/tex] such that [tex]\(C + B = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}\)[/tex]
To solve for [tex]\(C\)[/tex], we start from the equation:
[tex]\[ C + B = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix} \][/tex]
Subtract [tex]\( B \)[/tex] from both sides to isolate [tex]\( C \)[/tex]:
[tex]\[ C = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix} - B \][/tex]
Substitute [tex]\( B \)[/tex] into the equation:
[tex]\[ C = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix} - \begin{pmatrix}
-4 & -1 \\
-3 & -2
\end{pmatrix} \][/tex]
Subtract [tex]\( B \)[/tex] element-wise:
[tex]\[ C = \begin{pmatrix}
0 - (-4) & 0 - (-1) \\
0 - (-3) & 0 - (-2)
\end{pmatrix} = \begin{pmatrix}
4 & 1 \\
3 & 2
\end{pmatrix} \][/tex]
Therefore, the matrix [tex]\( C \)[/tex] that satisfies [tex]\(C + B = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}\)[/tex] is:
[tex]\[ C = \begin{pmatrix}
4 & 1 \\
3 & 2
\end{pmatrix} \][/tex]
### Final Summary
(a) The matrix [tex]\(2A + B\)[/tex] is:
[tex]\[ \begin{pmatrix}
-2 & 7 \\
1 & 4
\end{pmatrix} \][/tex]
(b) The matrix [tex]\(C\)[/tex] such that [tex]\(C + B = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}\)[/tex] is:
[tex]\[ \begin{pmatrix}
4 & 1 \\
3 & 2
\end{pmatrix} \][/tex]
