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$Consider this matrix: [[-3,6],[2,-5]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -3 & 6 \\ 2 & -5 \end{array}\right]$ $\newline$ 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}{cc} -3 & 6 \\ 2 & -5 \end{array}\right]

\newline

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

Write Matrix Elements: Write down the matrix and its elements. $\newline$ Matrix $A = \begin{bmatrix} -3 & 6 \ 2 & -5 \end{bmatrix}$ $\newline$ Let's denote the elements of the matrix as follows: $\newline$ $a = -3$ , $b = 6$ , $c = 2$ , $d = -5$
Calculate Determinant: Calculate the determinant of the matrix. $\newline$ The determinant of a $2$ x $2$ matrix $A$ is given by $ad - bc$ . $\newline$ $\text{det}(A) = (-3)(-5) - (6)(2)$ $\newline$ $\text{det}(A) = 15 - 12$ $\newline$ $\text{det}(A) = 3$
Check Non-Zero Determinant: Check if the determinant is non-zero. $\newline$ Since $\text{det}(A) = 3$ , which is non-zero, the matrix has an inverse.
Inverse Formula: Write down the formula for the inverse of a $2 \times 2$ matrix. $\newline$ The inverse of a $2 \times 2$ matrix $A$ is $(1/\text{det}(A)) \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$
Calculate Inverse: Plug the values into the formula to calculate the inverse. $\newline$ Inverse of $A = \frac{1}{3} \times \begin{bmatrix} -5 & -6 \ -2 & -3 \end{bmatrix}$
Simplify Inverse: Simplify the matrix by multiplying each element by the scalar $\frac{1}{3}$ . $\newline$ Inverse of $A = \left[\begin{array}{cc} -\frac{5}{3} & -\frac{6}{3} \ -\frac{2}{3} & -\frac{3}{3} \end{array}\right]$ $\newline$ Inverse of $A = \left[\begin{array}{cc} -\frac{5}{3} & -2 \ -\frac{2}{3} & -1 \end{array}\right]$