Question

find the inverse of the matrix \\(\

$$\begin{bmatrix} 1 & 2 & -2 \\\\ -1 & 3 & 0 \\\\ 0 & -2 & 1 \\end{bmatrix}$$

\\).

\\(\circ \

$$\begin{bmatrix} 3 & 2 & 6 \\\\ 1 & 1 & 2 \\\\ 2 & 2 & 5 \\end{bmatrix}$$

\\)

\\(\circ \

$$\begin{bmatrix} 1 & 2 & -2 \\\\ -1 & 3 & 0 \\\\ 0 & -2 & 1 \\end{bmatrix}$$

\\)

Explanation:

Calculate the determinant of the matrix

Using the Matrix Determinant knowledge point

$$ \det(A) = 1 \cdot (3 \cdot 1 - 0 \cdot (-2)) - 2 \cdot ((-1) \cdot 1 - 0 \cdot 0) + (-2) \cdot ((-1) \cdot (-2) - 3 \cdot 0) $$

$$ \det(A) = 1 \cdot 3 - 2 \cdot (-1) - 2 \cdot 2 = 3 + 2 - 4 = 1 $$

Find the cofactor matrix elements

Using the Cofactor Matrix knowledge point

$$
LATEXBLOCK0
$$

Construct the adjoint matrix

Using the Adjoint Matrix knowledge point

$$ \text{adj}(A) = C^T =
LATEXBLOCK1
$$

Compute the inverse matrix

Using the Matrix Inverse Formula knowledge point

$$ A^{-1} = \frac{1}{\det(A)} \text{adj}(A) = \frac{1}{1}
LATEXBLOCK2
=
LATEXBLOCK3
$$

Answer:

• **(A) \(

$$\begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}$$

\) (Correct answer)**

• (B) \(

$$\begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$$

\)