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$Consider this matrix: [[-9,-6],[0,2]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -9 & -6 \\ 0 & 2 \end{array}\right]$ $\newline$ 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} -9 & -6 \\ 0 & 2 \end{array}\right]

\newline

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

Calculate Determinant: To find the inverse of a $2 \times 2$ matrix given by $[[a, b], [c, d]]$ , we use the formula for the inverse of a $2 \times 2$ matrix: $\frac{1}{ad - bc} \times [[d, -b], [-c, a]]$ . First, we need to calculate the determinant of the matrix, which is $ad - bc$ .
Find Determinant: Calculate the determinant of the matrix $[[-9,-6],[0,2]]$ . The determinant is $(-9)(2) - (0)(-6) = -18 - 0 = -18$ .
Apply Inverse Formula: Since the determinant is not zero, the matrix has an inverse. Now, we apply the formula for the inverse of a $2 \times 2$ matrix. The inverse matrix is $\frac{1}{(-18)} \times \left[\begin{array}{cc} 2 & 6 \ 0 & -9 \end{array}\right]$ .
Multiply by $\frac{1}{(-18)}$ : Multiply each element of the matrix by $\frac{1}{(-18)}$ to get the inverse matrix. The resulting matrix is $\left[\begin{array}{cc} -\frac{2}{18} & -\frac{6}{18} \ 0 & \frac{9}{18} \end{array}\right]$ .
Simplify Fractions: Simplify the fractions in the matrix to get the final inverse matrix. The simplified matrix is $\left[\begin{array}{cc} \frac{1}{9} & -\frac{1}{3} \ 0 & -\frac{1}{2} \end{array}\right]$ .