Find the determinant of the matrix: \( A=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 3 & 2\end{array}\right] \)
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To find the determinant of the matrix \[ A=\begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 3 & 2 \end{bmatrix}, \] we can use the method of cofactor expansion along the first row. The determinant can be calculated as follows: \[ \text{det}(A) = 0 \cdot \text{det}(M_{11}) - 1 \cdot \text{det}(M_{12}) + 0 \cdot \text{det}(M_{13}), \] where \( M_{ij} \) is the minor matrix obtained by deleting the i-th row and j-th column. Here, we only have to compute \( \text{det}(M_{12}) \) since the first and third terms will be zero: \[ M_{12} = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}. \] The determinant of \( M_{12} \) is calculated as follows: \[ \text{det}(M_{12}) = (1)(2) - (1)(2) = 2 - 2 = 0. \] Now substituting this back, we have: \[ \text{det}(A) = 0 \cdot 0 - 1 \cdot 0 + 0 \cdot 0 = 0. \] Thus, the determinant of the matrix \( A \) is \[ \boxed{0}. \]
