To solve for matrix [tex]\(A\)[/tex] in the given matrix equation
[tex]\[
\left[\begin{array}{rr}2 & -1 \\ 2 & 0\end{array}\right] + 2A = \left[\begin{array}{rr}-3 & 5 \\ 4 & 3\end{array}\right],
\][/tex]
we perform the following steps:
1. Introduce Variables for Matrix [tex]\(A\)[/tex]:
Let [tex]\(A\)[/tex] be a matrix of unknowns:
[tex]\[
A = \left[\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right].
\][/tex]
2. Set Up the Matrix Equation:
Substitute [tex]\(A\)[/tex] into the given matrix equation:
[tex]\[
\left[\begin{array}{rr}2 & -1 \\ 2 & 0\end{array}\right] + 2 \left[\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right] = \left[\begin{array}{rr}-3 & 5 \\ 4 & 3\end{array}\right].
\][/tex]
3. Distribute the Scalar (2) over the Matrix [tex]\(A\)[/tex]:
[tex]\[
\left[\begin{array}{rr}2 & -1 \\ 2 & 0\end{array}\right] + \left[\begin{array}{cc}2a_{11} & 2a_{12} \\ 2a_{21} & 2a_{22}\end{array}\right] = \left[\begin{array}{rr}-3 & 5 \\ 4 & 3\end{array}\right].
\][/tex]
4. Combine the Matrices on the Left Side:
[tex]\[
\left[\begin{array}{cc}2 + 2a_{11} & -1 + 2a_{12} \\ 2 + 2a_{21} & 2a_{22}\end{array}\right] = \left[\begin{array}{rr}-3 & 5 \\ 4 & 3\end{array}\right].
\][/tex]
5. Set Up a System of Linear Equations:
By equating corresponding elements from both sides of the matrix equation, we obtain the following system of linear equations:
[tex]\[
\begin{cases}
2 + 2a_{11} = -3, \\
-1 + 2a_{12} = 5, \\
2 + 2a_{21} = 4, \\
2a_{22} = 3.
\end{cases}
\][/tex]
6. Solve the System of Equations:
- From the first equation:
[tex]\[
2 + 2a_{11} = -3 \implies 2a_{11} = -3 - 2 \implies 2a_{11} = -5 \implies a_{11} = -\frac{5}{2}.
\][/tex]
- From the second equation:
[tex]\[
-1 + 2a_{12} = 5 \implies 2a_{12} = 5 + 1 \implies 2a_{12} = 6 \implies a_{12} = 3.
\][/tex]
- From the third equation:
[tex]\[
2 + 2a_{21} = 4 \implies 2a_{21} = 4 - 2 \implies 2a_{21} = 2 \implies a_{21} = 1.
\][/tex]
- From the fourth equation:
[tex]\[
2a_{22} = 3 \implies a_{22} = \frac{3}{2}.
\][/tex]
7. Construct the Solution Matrix [tex]\(A\)[/tex]:
With the values found, we construct matrix [tex]\(A\)[/tex]:
[tex]\[
A = \left[\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right] = \left[\begin{array}{cc}-\frac{5}{2} & 3 \\ 1 & \frac{3}{2}\end{array}\right].
\][/tex]
Thus, the matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \left[\begin{array}{cc}-\frac{5}{2} & 3 \\ 1 & \frac{3}{2}\end{array}\right].
\][/tex]
