Step 1: Write down the matrix for which we want to find the determinant: \( A = \begin{pmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix} \).
Step 2: To find the determinant of a 3x3 matrix, we can use the formula: \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \), where the matrix is \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).
Step 3: Identify the elements of the matrix: \( a = 3, b = 1, c = 2, d = 1, e = 2, f = 3, g = 2, h = 3, i = 1 \).
Step 7: However, we notice that the rows of the matrix are linearly dependent (one row can be formed by a combination of the others), which means the determinant is actually 0.
Step 8: Therefore, the final answer is that the determinant is 0.
Determinants – The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.
Linear Dependence – Rows (or columns) of a matrix are linearly dependent if at least one row (or column) can be expressed as a linear combination of others, which results in a determinant of zero.
