$12$ . Find the matrix $X$ if, $\newline$ i) $\quad X\left[\begin{array}{cc}5 & 2 \\ -2 & 1\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 12 & 3\end{array}\right]$

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Determine the direction of a vector scalar multiple

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12

. Find the matrix

X

if,

\newline

\quad X\left[\begin{array}{cc}5 & 2 \\ -2 & 1\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 12 & 3\end{array}\right]

Set Up Equation: First, let's set up the equation with the given matrices. $X \cdot \begin{bmatrix} 5 & 2 \ -2 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 5 \ 12 & 3 \end{bmatrix}$
Find Inverse: To find $X$ , we need to multiply the inverse of $\left[\begin{array}{cc} 5 & 2 \ -2 & 1 \end{array}\right]$ with $\left[\begin{array}{cc} -1 & 5 \ 12 & 3 \end{array}\right]$ .
Calculate Determinant: Calculate the determinant of $\begin{bmatrix} 5 & 2 \ -2 & 1 \end{bmatrix}$ . $\newline$ $\text{det} = (5)(1) - (2)(-2) = 5 + 4 = 9$
Find Adjugate: Find the inverse of $\left[\begin{array}{cc}5 & 2 \ -2 & 1\end{array}\right]$ by dividing the adjugate matrix by the determinant. $\newline$ The adjugate matrix is $\left[\begin{array}{cc}1 & -2 \ 2 & 5\end{array}\right]$ . $\newline$ So, the inverse is $(\frac{1}{9}) * \left[\begin{array}{cc}1 & -2 \ 2 & 5\end{array}\right]$ .
Matrix Multiplication: Now, multiply the inverse of $\begin{bmatrix} 5 & 2 \ -2 & 1 \end{bmatrix}$ by $\begin{bmatrix} -1 & 5 \ 12 & 3 \end{bmatrix}$ . $X = \left(\frac{1}{9}\right) * \begin{bmatrix} 1 & -2 \ 2 & 5 \end{bmatrix} * \begin{bmatrix} -1 & 5 \ 12 & 3 \end{bmatrix}$
Perform Multiplication: Perform the matrix multiplication. $\newline$ $X = \frac{1}{9} \times \left[\begin{array}{cc} (1)(-1) + (-2)(12) & (1)(5) + (-2)(3) \ (2)(-1) + (5)(12) & (2)(5) + (5)(3) \end{array}\right]$ $\newline$ $X = \frac{1}{9} \times \left[\begin{array}{cc} -1 - 24 & 5 - 6 \ -2 + 60 & 10 + 15 \end{array}\right]$
Simplify Result: Simplify the matrix multiplication. $X = \frac{1}{9} \times \begin{bmatrix} -25 & -1 \ 58 & 25 \end{bmatrix}$
Final Matrix: Divide each element by $9$ to get the final matrix $X$ . $X = \left[\begin{array}{cc}-\frac{25}{9} & -\frac{1}{9} \ \frac{58}{9} & \frac{25}{9}\end{array}\right]$