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Solve each equation and check your answers.

[tex]\[ \begin{pmatrix} 2 & 1 \\ 3 & -4 \end{pmatrix} - 3X = \begin{pmatrix} 9 & -67 \\ 1 & -2 \end{pmatrix} \][/tex]

Sagot :

GinnyAnsweridn
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]
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