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$Consider this matrix: [[10,0],[-8,1]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} 10 & 0 \\ -8 & 1 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

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Math Problems

Algebra 2

Find inverse functions and relations

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Q. Consider this matrix:

\newline

\left[\begin{array}{cc} 10 & 0 \\ -8 & 1 \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 $2$ x $2$ matrix, we use the formula: $\newline$ $A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ $\newline$ where $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\text{det}(A) = ad - bc$ . $\newline$ For the given matrix $A = \begin{bmatrix} 10 & 0 \\ -8 & 1 \end{bmatrix}$ , we have $a = 10$ , $b = 0$ , $c = -8$ , and $d = 1$ . $\newline$ First, we calculate the determinant of $A$ : $\newline$ $\text{det}(A) = (10)(1) - (0)(-8) = 10 - 0 = 10$ .
Apply Inverse Formula: Now that we have the determinant, we can find the inverse by applying the formula: $\newline$ $A^{-1} = \frac{1}{10} \begin{bmatrix} 1 & -0 \\ 8 & 10 \end{bmatrix}$ $\newline$ Simplify the matrix by multiplying each element by $\frac{1}{10}$ : $\newline$ $A^{-1} = \begin{bmatrix} \frac{1}{10} & 0 \\ \frac{8}{10} & 1 \end{bmatrix}$ $\newline$ Further simplification gives us: $\newline$ $A^{-1} = \begin{bmatrix} 0.1 & 0 \\ 0.8 & 1 \end{bmatrix}$