Consider this matrix: [[-10,-10],[6,9]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Question

Consider this matrix:  [[-10,-10],[6,9]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Bytelearn · Accepted Answer

Formula for Inverse: To find the inverse of a 2x2 matrix, we use the formula: $$ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $$ where $ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} $. For the given matrix $ A = \begin{bmatrix} -10 & -10 \ 6 & 9 \end{bmatrix} $, we have $ a = -10 $, $ b = -10 $, $ c = 6 $, and $ d = 9 $. First, we need to calculate the determinant of $ A $, which is $ \text{det}(A) = ad - bc $.Calculate Determinant: Calculate the determinant: $$ \text{det}(A) = (-10)(9) - (-10)(6) $$ $$ \text{det}(A) = -90 - (-60) $$ $$ \text{det}(A) = -90 + 60 $$ $$ \text{det}(A) = -30 $$ The determinant is -30.Apply Inverse Formula: Now, we apply the formula for the inverse of a 2x2 matrix: $$ A^{-1} = \frac{1}{-30} \begin{bmatrix} 9 & 10 \ -6 & -10 \end{bmatrix} $$ $$ A^{-1} = \begin{bmatrix} -9/30 & -10/30 \ 6/30 & 10/30 \end{bmatrix} $$ Simplify the fractions: $$ A^{-1} = \begin{bmatrix} -3/10 & -1/3 \ 1/5 & 1/3 \end{bmatrix} $$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -10 & -10 \\ 6 & 9 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Full solution

More problems from Find inverse functions and relations

Related topics

Consider this matrix:\newline[−10amp;−106amp;9] \left[\begin{array}{cc} -10 &amp; -10 \\ 6 &amp; 9 \end{array}\right] [−106​amp;−10amp;9​]\newlineFind the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Consider this matrix: $\newline$ $\left[\begin{array}{cc} -10 & -10 \\ 6 & 9 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.