To solve for the unknown matrix [tex]\(X\)[/tex] in the equation
[tex]\[
\left[\begin{array}{cc}
2 & 1 \\
3 & -4
\end{array}\right] - 3X = \left[\begin{array}{cc}
9 & -67 \\
1 & -2
\end{array}\right],
\][/tex]
we'll follow these steps:
### Step 1: Write the Given Equation in Matrix Form
The equation given is:
[tex]\[
A - 3X = B,
\][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are known matrices:
[tex]\[
A = \left[\begin{array}{cc}
2 & 1 \\
3 & -4
\end{array}\right]
\][/tex]
[tex]\[
B = \left[\begin{array}{cc}
9 & -67 \\
1 & -2
\end{array}\right].
\][/tex]
### Step 2: Isolate [tex]\(3X\)[/tex]
We isolate [tex]\(3X\)[/tex] by subtracting matrix [tex]\(B\)[/tex] from matrix [tex]\(A\)[/tex]:
[tex]\[
A - B = 3X.
\][/tex]
Since matrix subtraction is done element-wise, let's compute [tex]\(A - B\)[/tex]:
[tex]\[
A - B = \left[\begin{array}{cc}
2 & 1 \\
3 & -4
\end{array}\right] - \left[\begin{array}{cc}
9 & -67 \\
1 & -2
\end{array}\right].
\][/tex]
### Step 3: Compute the Difference Matrix
Subtract the corresponding elements of matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A - B = \left[\begin{array}{cc}
2 - 9 & 1 - (-67) \\
3 - 1 & -4 - (-2)
\end{array}\right] = \left[\begin{array}{cc}
-7 & 68 \\
2 & -2
\end{array}\right].
\][/tex]
So, the difference matrix is:
[tex]\[
A - B = \left[\begin{array}{cc}
-7 & 68 \\
2 & -2
\end{array}\right].
\][/tex]
### Step 4: Solve for [tex]\(X\)[/tex]
We have the equation:
[tex]\[
3X = \left[\begin{array}{cc}
-7 & 68 \\
2 & -2
\end{array}\right].
\][/tex]
To solve for [tex]\(X\)[/tex], we divide each element of the resulting matrix [tex]\(A - B\)[/tex] by 3:
[tex]\[
X = \frac{1}{3} \left[\begin{array}{cc}
-7 & 68 \\
2 & -2
\end{array}\right] = \left[\begin{array}{cc}
\frac{-7}{3} & \frac{68}{3} \\
\frac{2}{3} & \frac{-2}{3}
\end{array}\right].
\][/tex]
Simplify the elements:
[tex]\[
X = \left[\begin{array}{cc}
-2.33333333 & 22.66666667 \\
0.66666667 & -0.66666667
\end{array}\right].
\][/tex]
Thus, the unknown matrix [tex]\(X\)[/tex] is:
[tex]\[
X = \left[\begin{array}{cc}
-2.33333333 & 22.66666667 \\
0.66666667 & -0.66666667
\end{array}\right].
\][/tex]
### Final Answer
The difference matrix is:
[tex]\[
\left[\begin{array}{cc}
-7 & 68 \\
2 & -2
\end{array}\right],
\][/tex]
and the unknown matrix [tex]\(X\)[/tex] is:
[tex]\[
\left[\begin{array}{cc}
-2.33333333 & 22.66666667 \\
0.66666667 & -0.66666667
\end{array}\right].
\][/tex]
