Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
