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Sagot :
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]
[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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