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$Consider this matrix: [[5,-3],[6,-3]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{ll} 5 & -3 \\ 6 & -3 \end{array}\right]$ $\newline$ 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}{ll} 5 & -3 \\ 6 & -3 \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.
Find Adjugate: The determinant of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is calculated as $ad - bc$ . For our matrix $\left[\begin{array}{cc} 5 & -3 \ 6 & -3 \end{array}\right]$ , the determinant is $(5 \times -3) - (6 \times -3)$ .
Calculate Inverse: Calculating the determinant: $\newline$ det $(A) = (5 \times -3) - (6 \times -3)$ $\newline$ det $(A) = -15 - (-18)$ $\newline$ det $(A) = -15 + 18$ $\newline$ det $(A) = 3$ $\newline$ The determinant of the matrix is $3$ .
Multiply by $1$ / $3$ : Next, we need to find the adjugate of the matrix. The adjugate of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $\left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ . For our matrix $\left[\begin{array}{cc} 5 & -3 \ 6 & -3 \end{array}\right]$ , the adjugate is $\left[\begin{array}{cc} -3 & 3 \ -6 & 5 \end{array}\right]$ .
Simplify Fraction: Now we can find the inverse of the matrix by multiplying the adjugate by $\frac{1}{\det(A)}$ . $\newline$ Inverse $(A) = \frac{1}{3} \times \begin{bmatrix} -3 & 3 \ -6 & 5 \end{bmatrix}$
Simplify Fraction: Now we can find the inverse of the matrix by multiplying the adjugate by $1/\text{det}(A)$ . $\newline$ Inverse $(A) = (1/3) \times \left[\begin{array}{cc} -3 & 3 \ -6 & 5 \end{array}\right]$ Multiplying the adjugate by $1/3$ : $\newline$ Inverse $(A) = \left[\begin{array}{cc} (-3/3) & (3/3) \ (-6/3) & (5/3) \end{array}\right]$ $\newline$ Inverse $(A) = \left[\begin{array}{cc} -1 & 1 \ -2 & 5/3 \end{array}\right]$
Simplify Fraction: Now we can find the inverse of the matrix by multiplying the adjugate by $1/\text{det}(A)$ . $\newline$ Inverse $(A) = (1/3) \times \left[\begin{array}{cc} -3 & 3 \ -6 & 5 \end{array}\right]$ Multiplying the adjugate by $1/3$ : $\newline$ Inverse $(A) = \left[\begin{array}{cc} (-3/3) & (3/3) \ (-6/3) & (5/3) \end{array}\right]$ $\newline$ Inverse $(A) = \left[\begin{array}{cc} -1 & 1 \ -2 & 5/3 \end{array}\right]$ We can simplify the fraction $5/3$ to its decimal equivalent if needed. $\newline$ $5/3$ is approximately $1.6667$ . $\newline$ So the inverse matrix can also be written as: $\newline$ Inverse $(A) = \left[\begin{array}{cc} -1 & 1 \ -2 & 1.6667 \end{array}\right]$