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$Consider this matrix: [[-10,1],[2,-1]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -10 & 1 \\ 2 & -1 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

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Q. Consider this matrix:

\newline

\left[\begin{array}{cc} -10 & 1 \\ 2 & -1 \end{array}\right]

\newline

Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Write Matrix Determinant: Write down the matrix and its determinant. $\newline$ The determinant of a $2 \times 2$ matrix $\left[\begin{array}{cc}a & b \ c & d\end{array}\right]$ is $ad - bc$ . $\newline$ For the matrix $\left[\begin{array}{cc}-10 & 1 \ 2 & -1\end{array}\right]$ , the determinant is $(-10)(-1) - (1)(2)$ .
Calculate Determinant: Calculate the determinant. $\newline$ Determinant = $(-10)(-1) - (1)(2) = 10 - 2 = 8$ .
Find Inverse Matrix: Find the inverse of the matrix. $\newline$ The inverse of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $\frac{1}{\text{determinant}} \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ . $\newline$ For our matrix, the inverse will be $\frac{1}{8} \times \left[\begin{array}{cc} -1 & -1 \ -2 & -10 \end{array}\right]$ .
Multiply by Reciprocal: Multiply the matrix by the reciprocal of the determinant. $\newline$ Inverse matrix = $(\frac{1}{8}) \times \left[\begin{array}{cc} -1 & -1 \ -2 & -10 \end{array}\right] = \left[\begin{array}{cc} -\frac{1}{8} & -\frac{1}{8} \ -\frac{2}{8} & -\frac{10}{8} \end{array}\right]$ .
Simplify Fractions: Simplify the fractions in the inverse matrix. $\newline$ Inverse matrix = \left[\begin{array}{cc}$\newline-\frac{1}{8} & -\frac{1}{8} (\newline$-\frac{1}{4} & -\frac{5}{4} $\newline$ \end{array}\right]\).