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$Consider this matrix: [[-5,7],[-1,2]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{ll} -5 & 7 \\ -1 & 2 \end{array}\right]$ $\newline$ 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}{ll} -5 & 7 \\ -1 & 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, we use the formula: $\newline$ $\text{Inverse}(A) = \left(\frac{1}{\text{det}(A)}\right) \times \text{adj}(A)$ $\newline$ where $\text{det}(A)$ is the determinant of matrix $A$ and $\text{adj}(A)$ is the adjugate of matrix $A$ . $\newline$ First, we need to calculate the determinant of the given matrix. $\newline$ The determinant of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is calculated as $ad - bc$ . $\newline$ For our matrix, $\text{det}(A) = (-5)(2) - (7)(-1) = -10 + 7 = -3$ .
Find Adjugate: Now that we have the determinant, we can find the adjugate of the matrix. $\newline$ The adjugate of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $\left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ . $\newline$ For our matrix, $\text{adj}(A) = \left[\begin{array}{cc} 2 & -7 \ 1 & -5 \end{array}\right]$ .
Calculate Inverse: Next, we multiply the adjugate by the reciprocal of the determinant to find the inverse. $\newline$ So, the inverse of our matrix is $(1/-3) \times [[2, -7], [1, -5]]$ . $\newline$ This gives us the matrix $[[-2/3, 7/3], [-1/3, 5/3]]$ .