Solution: The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).
Steps: 9
Step 1: Write down the determinant you need to evaluate: \( \begin{vmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{vmatrix} \).
Step 2: Identify the first row of the matrix, which is (1, 2, 1).
Step 3: Use the formula for the determinant of a 3x3 matrix: \( a(ei - fh) - b(di - fg) + c(dh - eg) \), where a, b, c are the elements of the first row, and d, e, f, g, h, i are the elements of the remaining rows.
Step 4: Assign values: a = 1, b = 2, c = 1, d = 0, e = 1, f = 0, g = 2, h = 3, i = 1.
Step 5: Calculate the first part: \( 1(1*1 - 0*3) = 1(1 - 0) = 1 \).
Step 6: Calculate the second part: \( -2(0*1 - 0*2) = -2(0 - 0) = -2(0) = 0 \).
Step 7: Calculate the third part: \( 1(0*3 - 1*2) = 1(0 - 2) = 1(-2) = -2 \).
Step 8: Combine all parts: \( 1 + 0 - 2 = -1 \).
Step 9: The final result of the determinant is \( -1 \).
