Parry Mitchell
07/08/2024 · Junior High School
If \( A=\left[\begin{array}{ll}3 & -1 \\ 2 & -4\end{array}\right], B=\left[\begin{array}{cc}1 & 2 \\ -3 & 0\end{array}\right] \) then find the matrix ' \( X \) ' such that \( 2 X+3 A-4 B=1 \) where, 1 is identity matrix
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To find the matrix \( X \) such that \( 2X + 3A - 4B = I \), where \( I \) is the identity matrix, we first need to compute \( 3A \) and \( -4B \).
Given:
\[
A = \begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix}
\]
First, we calculate \( 3A \):
\[
3A = 3 \begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix} = \begin{bmatrix} 9 & -3 \\ 6 & -12 \end{bmatrix}
\]
Next, we calculate \( -4B \):
\[
-4B = -4 \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} = \begin{bmatrix} -4 & -8 \\ 12 & 0 \end{bmatrix}
\]
Now, we can find \( 3A - 4B \):
\[
3A - 4B = \begin{bmatrix} 9 & -3 \\ 6 & -12 \end{bmatrix} + \begin{bmatrix} -4 & -8 \\ 12 & 0 \end{bmatrix} = \begin{bmatrix} 9 - 4 & -3 - 8 \\ 6 + 12 & -12 + 0 \end{bmatrix} = \begin{bmatrix} 5 & -11 \\ 18 & -12 \end{bmatrix}
\]
Now we have:
\[
2X + 3A - 4B = I \implies 2X + \begin{bmatrix} 5 & -11 \\ 18 & -12 \end{bmatrix} = I
\]
The identity matrix \( I \) for a \( 2 \times 2 \) matrix is:
\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]
Now we can set up the equation:
\[
2X + \begin{bmatrix} 5 & -11 \\ 18 & -12 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]
Subtract \( \begin{bmatrix} 5 & -11 \\ 18 & -12 \end{bmatrix} \) from both sides:
\[
2X = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 5 & -11 \\ 18 & -12 \end{bmatrix} = \begin{bmatrix} 1 - 5 & 0 + 11 \\ 0 - 18 & 1 + 12 \end{bmatrix} = \begin{bmatrix} -4 & 11 \\ -18 & 13 \end{bmatrix}
\]
Now, divide both sides by 2 to solve for \( X \):
\[
X = \frac{1}{2} \begin{bmatrix} -4 & 11 \\ -18 & 13 \end{bmatrix} = \begin{bmatrix} -2 & \frac{11}{2} \\ -9 & \frac{13}{2} \end{bmatrix}
\]
Thus, the matrix \( X \) is:
\[
\boxed{\begin{bmatrix} -2 & \frac{11}{2} \\ -9 & \frac{13}{2} \end{bmatrix}}
\]
Quick Answer
The matrix \( X \) is:
\[
\boxed{\begin{bmatrix} -2 & \frac{11}{2} \\ -9 & \frac{13}{2} \end{bmatrix}}
\]
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