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$Consider this matrix: [[-4,0],[-10,-6]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -4 & 0 \\ & \\ -10 & -6 \end{array}\right]$ $\newline$ 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} -4 & 0 \\ & \\ -10 & -6 \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 $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ , we use the formula for the inverse of a $2 \times 2$ matrix: $\frac{1}{ad - bc} \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ . First, we need to calculate the determinant $ad - bc$ .
Determinant calculation: The determinant of our matrix $[[-4,0],[-10,-6]]$ is calculated as $(-4)\cdot(-6) - (0)\cdot(-10)$ .
Find inverse matrix: The determinant is $(-4)\cdot(-6) - (0)\cdot(-10) = 24 - 0 = 24$ .
Multiply by reciprocal: Now that we have the determinant, we can find the inverse by multiplying the reciprocal of the determinant by the matrix $[[d, -b], [-c, a]]$ . For our matrix, $a = -4$ , $b = 0$ , $c = -10$ , and $d = -6$ . So the matrix $[[d, -b], [-c, a]]$ becomes $[[-6, 0], [10, -4]]$ .
Calculate inverse matrix: Multiplying the reciprocal of the determinant, which is $\frac{1}{24}$ , by the matrix $\left[\begin{array}{cc} -6 & 0 \ 10 & -4 \end{array}\right]$ gives us the inverse matrix. We perform the multiplication for each element of the matrix.
Simplify fractions: The inverse matrix is $\frac{1}{24} \times \left[\begin{array}{cc} -6 & 0 \ 10 & -4 \end{array}\right]$ which simplifies to $\left[\begin{array}{cc} \frac{-6}{24} & \frac{0}{24} \ \frac{10}{24} & \frac{-4}{24} \end{array}\right]$ .
Final inverse matrix: Simplify the fractions in the matrix to get $\left[\begin{array}{cc} -\frac{1}{4} & 0 \ \frac{5}{12} & -\frac{1}{6} \end{array}\right]$ . This is the inverse of the original matrix.