Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 &am

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

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

Correct Answer: 10

Step 1: Identify the matrix. We have the matrix A = \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
Step 2: Write down the formula for the determinant of a 3x3 matrix. The formula is: \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \), where the matrix is \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).
Step 3: Assign values from the matrix to the variables in the formula. Here, a = 2, b = 3, c = 1, d = 1, e = 0, f = 2, g = 4, h = 1, i = 0.
Step 4: Calculate the parts of the formula. First, calculate ei - fh: 0*0 - 2*1 = 0 - 2 = -2.
Step 5: Calculate di - fg: 1*0 - 2*4 = 0 - 8 = -8.
Step 6: Calculate dh - eg: 1*1 - 0*4 = 1 - 0 = 1.
Step 7: Substitute these values back into the determinant formula: \( \text{det}(A) = 2(-2) - 3(-8) + 1(1) \).
Step 8: Simplify the expression: \( \text{det}(A) = -4 + 24 + 1 = 21 \).
Step 9: Therefore, the determinant of the matrix is 21.

Determinant Calculation – The question tests the ability to compute the determinant of a 3x3 matrix using the appropriate formula.
Matrix Properties – Understanding properties of matrices, such as row operations and their effect on the determinant.