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

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

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: 10

Step 1: Identify the matrix C. It is given as C = [[2, 1, 3], [1, 0, 2], [3, 2, 1]].
Step 2: Write down the formula for the determinant of a 3x3 matrix. The formula is: |C| = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is: [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign the values from matrix C to the variables in the formula: a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 3, h = 2, i = 1.
Step 4: Calculate the first part of the formula: ei - fh = (0*1) - (2*2) = 0 - 4 = -4.
Step 5: Calculate the second part of the formula: di - fg = (1*1) - (2*3) = 1 - 6 = -5.
Step 6: Calculate the third part of the formula: dh - eg = (1*2) - (0*3) = 2 - 0 = 2.
Step 7: Substitute these values back into the determinant formula: |C| = 2(-4) - 1(-5) + 3(2).
Step 8: Calculate each term: 2(-4) = -8, -1(-5) = 5, and 3(2) = 6.
Step 9: Add these results together: -8 + 5 + 6 = 3.
Step 10: Conclude that the determinant |C| = 3.