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$Consider this matrix: [[-10,-10],[6,9]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -10 & -10 \\ 6 & 9 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

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Algebra 2

Find inverse functions and relations

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Q. Consider this matrix:

\newline

\left[\begin{array}{cc} -10 & -10 \\ 6 & 9 \end{array}\right]

\newline

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

Formula for Inverse: To find the inverse of a $2$ x $2$ matrix, we use the formula: $\newline$ $A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ $\newline$ where $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ . $\newline$ For the given matrix $A = \begin{bmatrix} -10 & -10 \\ 6 & 9 \end{bmatrix}$ , we have $a = -10$ , $b = -10$ , $c = 6$ , and $d = 9$ . $\newline$ First, we need to calculate the determinant of $A$ , which is $\text{det}(A) = ad - bc$ .
Calculate Determinant: Calculate the determinant: $\newline$ $\text{det}(A) = (-10)(9) - (-10)(6)$ $\newline$ $\text{det}(A) = -90 - (-60)$ $\newline$ $\text{det}(A) = -90 + 60$ $\newline$ $\text{det}(A) = -30$ $\newline$ The determinant is $-30$ .
Apply Inverse Formula: Now, we apply the formula for the inverse of a $2$ x $2$ matrix: $\newline$ $A^{-1} = \frac{1}{-30} \begin{bmatrix} 9 & 10 \\ -6 & -10 \end{bmatrix}$ $\newline$ $A^{-1} = \begin{bmatrix} -9/30 & -10/30 \\ 6/30 & 10/30 \end{bmatrix}$ $\newline$ Simplify the fractions: $\newline$ $A^{-1} = \begin{bmatrix} -3/10 & -1/3 \\ 1/5 & 1/3 \end{bmatrix}$