Evaluate the determinant \( \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).

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Q: Evaluate the determinant \( \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).

Solution: The determinant is calculated as \( 2(0*1 - 2*2) - 1(1*1 - 2*3) + 3(1*2 - 0*3) = 2(0 - 4) - 1(1 - 6) + 3(2) = -8 + 5 + 6 = 3 \).

Steps: 9

Step 1: Write down the matrix: \( A = \begin{pmatrix} 2 & 1 & 3 \ 1 & 0 & 2 \ 3 & 2 & 1 \ end{pmatrix} \).
Step 2: Identify the elements of the matrix: \( a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 3, h = 2, i = 1 \).
Step 3: Use the formula for the determinant of a 3x3 matrix: \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
Step 4: Substitute the values into the formula: \( \text{det}(A) = 2(0*1 - 2*2) - 1(1*1 - 2*3) + 3(1*2 - 0*3) \).
Step 5: Calculate each part: \( 0*1 - 2*2 = 0 - 4 = -4 \), \( 1*1 - 2*3 = 1 - 6 = -5 \), and \( 1*2 - 0*3 = 2 - 0 = 2 \).
Step 6: Substitute these results back into the determinant formula: \( \text{det}(A) = 2(-4) - 1(-5) + 3(2) \).
Step 7: Calculate each term: \( 2(-4) = -8 \), \( -1(-5) = 5 \), and \( 3(2) = 6 \).
Step 8: Add the results together: \( -8 + 5 + 6 = -8 + 11 = 3 \).
Step 9: Conclude that the determinant of the matrix is \( 3 \).