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$Consider this matrix: [[3,6],[-10,-10]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} 3 & 6 \\ & \\ -10 & -10 \end{array}\right]$ $\newline$ 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 \\ & \\ -10 & -10 \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 Determinant: Write down the matrix and its determinant. $\newline$ The determinant of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $ad - bc$ . $\newline$ For the matrix $\left[\begin{array}{cc} 3 & 6 \ -10 & -10 \end{array}\right]$ , the determinant is $(3 \times -10) - (6 \times -10)$ .
Calculate Determinant: Calculate the determinant. $\newline$ Determinant = $(3 \times -10) - (6 \times -10) = -30 - (-60) = -30 + 60 = 30$ .
Check Non-Zero Determinant: Check if the determinant is non-zero. $\newline$ Since the determinant is $30$ , which is non-zero, the matrix has an inverse.
Write Inverse Formula: Write down the formula for the inverse of a $2 \times 2$ matrix. $\newline$ The inverse of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $\frac{1}{\text{determinant}} \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ .
Apply Inverse Formula: Apply the formula to find the inverse of the given matrix. $\newline$ Using the formula, the inverse of \left[\begin{array}{cc}3 & 6 \-10 & -10\end{array}\right] is $\frac{1}{30}$ * \left[\begin{array}{cc}-10 & -6 \10 & 3\end{array}\right].
Multiply by Scalar: Multiply each element of the matrix by the scalar $(1/30)$ . $\newline$ The inverse matrix is $[[-10/30, -6/30], [10/30, 3/30]]$ .
Simplify Inverse Matrix: Simplify the fractions in the inverse matrix. $\newline$ The simplified inverse matrix is $\left[\begin{array}{cc} -\frac{1}{3} & -\frac{1}{5} \ \frac{1}{3} & \frac{1}{10} \end{array}\right]$ .