QuestionFind the inverse of the matrix. $$\left[\begin{array}{ccc} -1 & -1 & 2 \\ 3 & 2 & -4 \\ -0.5 & 0 & 0.5 \end{array}\right]$$ - $\left[\begin{array}{lll}2 & 1 & 0 \\ 1 & 1 & 4 \\ 2 & 1 & 2\end{array}\right]$ $\left[\begin{array}{lll}2 & 0 & 4 \\ 1 & 0 & 1 \\ 2 & 2 & 2\end{array}\right]$ $\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 1 & 4 \\ 2 & 1 & 1\end{array}\right]$ $\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 4 & 2 \\ 1 & 1 & 2\end{array}\right]$

Studdy Solution
Calculate the inverse using the formula $A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$:
$A^{-1} = \frac{1}{1.5} \begin{bmatrix} 1 & -0.5 & -2 \\ 0.5 & 0.5 & -2 \\ 1 & 0.5 & 1 \end{bmatrix}$
$= \begin{bmatrix} \frac{1}{1.5} & \frac{-0.5}{1.5} & \frac{-2}{1.5} \\ \frac{0.5}{1.5} & \frac{0.5}{1.5} & \frac{-2}{1.5} \\ \frac{1}{1.5} & \frac{0.5}{1.5} & \frac{1}{1.5} \end{bmatrix}$
$= \begin{bmatrix} \frac{2}{3} & \frac{-1}{3} & \frac{-4}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{-4}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \end{bmatrix}$
The inverse of the matrix is:
$\boxed{\begin{bmatrix} \frac{2}{3} & \frac{-1}{3} & \frac{-4}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{-4}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \end{bmatrix}}$