Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).

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Q: Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).

Solution: The determinant evaluates to 0.

Steps: 8

Step 1: Write down the matrix: \( A = \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
Step 2: Use the formula for the determinant of a 3x3 matrix: \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \), where the matrix is \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).
Step 3: Identify the elements from the matrix: \( a = 2, b = 3, c = 1, d = 1, e = 0, f = 4, g = 5, h = 2, i = 1 \).
Step 4: Calculate the products: \( ei - fh = 0 \cdot 1 - 4 \cdot 2 = 0 - 8 = -8 \), \( di - fg = 1 \cdot 1 - 4 \cdot 5 = 1 - 20 = -19 \), and \( dh - eg = 1 \cdot 2 - 0 \cdot 5 = 2 - 0 = 2 \).
Step 5: Substitute these values into the determinant formula: \( \text{det}(A) = 2(-8) - 3(-19) + 1(2) \).
Step 6: Calculate each term: \( 2(-8) = -16 \), \( -3(-19) = 57 \), and \( 1(2) = 2 \).
Step 7: Add these results together: \( -16 + 57 + 2 = 43 \).
Step 8: Conclude that the determinant of the matrix is 43.