Find the deteominants \[ \left[\begin{array}{ccc}2 & -1 & 1 \\ 3 & 1 & 2 \\ 1 & 1 & 1\end{array}\right] \]

To find the determinant of the matrix \[ A = \begin{bmatrix} 2 & -1 & 1 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}, \] we can use the formula for the determinant of a 3x3 matrix: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \] For our matrix, we assign the values as follows: \[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = \begin{bmatrix} 2 & -1 & 1 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}, \] where \( a = 2, b = -1, c = 1, d = 3, e = 1, f = 2, g = 1, h = 1, i = 1 \). Now, we can plug these values into the determinant formula: 1. For \( ei - fh = 1 \cdot 1 - 2 \cdot 1 = 1 - 2 = -1 \). 2. For \( di - fg = 3 \cdot 1 - 2 \cdot 1 = 3 - 2 = 1 \). 3. For \( dh - eg = 3 \cdot 1 - 1 \cdot 1 = 3 - 1 = 2 \). Now substituting these results back into the determinant formula: \[ \text{det}(A) = 2(-1) - (-1)(1) + 1(2) \] \[ = -2 + 1 + 2 \] \[ = -2 + 3 = 1. \] Thus, the determinant of the matrix is \[ \boxed{1}. \]