Evaluate \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6

Evaluate \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \)

Evaluate \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \)

Correct Answer: 1

Step 1: Identify the matrix for which we need to calculate the determinant: \( A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \).
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: Assign values from the matrix: \( a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0 \).
Step 4: Calculate the first part: \( a(ei - fh) = 1(1*0 - 4*6) = 1(0 - 24) = -24 \).
Step 5: Calculate the second part: \( -b(di - fg) = -2(0*0 - 4*5) = -2(0 - 20) = 40 \).
Step 6: Calculate the third part: \( c(dh - eg) = 3(0*6 - 1*5) = 3(0 - 5) = -15 \).
Step 7: Combine all parts: \( -24 + 40 - 15 = 1 \).
Step 8: Conclude that the determinant of the matrix is \( 1 \).

Determinants – The question tests the ability to calculate the determinant of a 3x3 matrix using the cofactor expansion method.
Cofactor Expansion – The solution involves applying the cofactor expansion along the first row of the matrix.