Matrix Determinant Calculation: |B| = |A|adj(A)

The objective is to find the determinant of matrix $B$, defined as $B = |A|\text{adj}(A)$. The given matrix $A$ is a $3 \times 3$ matrix. We need to utilize properties of determinants and adjoints.

1. Calculate Determinant |A|

Compute the determinant of the matrix $A = \begin{bmatrix} 2 & 1 & -2 \\ 1 & 1 & -1 \\ 1 & 0 & 2 \end{bmatrix}$.

The determinant $|A|$ is calculated as:

$|A| = 2 \begin{vmatrix} 1 & -1 \\ 0 & 2 \end{vmatrix} - 1 \begin{vmatrix} 1 & -1 \\ 1 & 2 \end{vmatrix} + (-2) \begin{vmatrix} 1 & 1 \\ 1 & 0 \end{vmatrix}$ $|A| = 2( (1)(2) - (-1)(0) ) - 1( (1)(2) - (-1)(1) ) - 2( (1)(0) - (1)(1) )$ $|A| = 2(2 - 0) - 1(2 + 1) - 2(0 - 1)$ $|A| = 2(2) - 1(3) - 2(-1)$ $|A| = 4 - 3 + 2 = 3$

2. Calculate Adjoint Determinant |adj(A)|

Use the property $|\text{adj}(A)| = |A|^{n-1}$ for an $n \times n$ matrix $A$. In this problem, the matrix $A$ is $3 \times 3$, so $n=3$. With $|A| = 3$, we get:

$|\text{adj}(A)| = |A|^{3-1} = |A|^2$ $|\text{adj}(A)| = 3^2 = 9$

3. Calculate Matrix B Determinant |B|

We are given $B = |A|\text{adj}(A)$. Since $|A|=3$, this simplifies to $B = 3 \cdot \text{adj}(A)$.

To find the determinant $|B|$, we apply the scalar multiplication property of determinants: $|kM| = k^n |M|$, where $k$ is a scalar and $M$ is an $n \times n$ matrix.

$|B| = |3 \cdot \text{adj}(A)|$

Here, the scalar $k=3$ and the matrix is $\text{adj}(A)$, which is also a $3 \times 3$ matrix ($n=3$).

$|B| = 3^3 |\text{adj}(A)|$ $|B| = 27 \cdot |\text{adj}(A)|$

Substitute the value of $|\text{adj}(A)|$ calculated in Step 2:

$|B| = 27 \cdot 9 = 243$