To solve for the matrix [tex]\(C\)[/tex] where [tex]\(C = A - B\)[/tex], we will perform element-wise subtraction of matrix [tex]\(B\)[/tex] from matrix [tex]\(A\)[/tex].
Given:
[tex]\[
A = \begin{pmatrix}
2 & -3 \\
-1 & 4
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
4 & -5 \\
1 & 3
\end{pmatrix}
\][/tex]
Step-by-Step Solution:
1. Subtract the corresponding elements of matrix [tex]\(B\)[/tex] from matrix [tex]\(A\)[/tex] to find matrix [tex]\(C\)[/tex]:
[tex]\[
C = A - B = \begin{pmatrix}
2 & -3 \\
-1 & 4
\end{pmatrix} - \begin{pmatrix}
4 & -5 \\
1 & 3
\end{pmatrix}
\][/tex]
2. Perform the element-wise subtraction:
- For element [tex]\(C_{1,1}\)[/tex]:
[tex]\[ 2 - 4 = -2 \][/tex]
- For element [tex]\(C_{1,2}\)[/tex]:
[tex]\[ -3 - (-5) = -3 + 5 = 2 \][/tex]
- For element [tex]\(C_{2,1}\)[/tex]:
[tex]\[ -1 - 1 = -2 \][/tex]
- For element [tex]\(C_{2,2}\)[/tex]:
[tex]\[ 4 - 3 = 1 \][/tex]
3. Thus, the resulting matrix [tex]\(C\)[/tex] is:
[tex]\[
C = \begin{pmatrix}
-2 & 2 \\
-2 & 1
\end{pmatrix}
\][/tex]
4. To find the element [tex]\(c_{1,2}\)[/tex], which refers to the element in the first row and second column of matrix [tex]\(C\)[/tex]:
[tex]\[
c_{1,2} = 2
\][/tex]
Therefore, the element [tex]\(c_{1,2}\)[/tex] is [tex]\(2\)[/tex].
