Question

The value of \[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix} \textbf{is:} \]

• A. \( 0 \)
• B. \( 2 \)
• C. \( 7 \)
• D. \( -2 \)

Hint:

To calculate the determinant of a \( 3 \times 3 \) matrix, expand along any row or column and simplify using \( 2 \times 2 \) minors.

Answer 1

The Correct Option is A

Step 1: Formulate the determinant.
The determinant to be calculated is: \[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix}. \] Step 2: Expand the determinant along the first row.
The expansion along the first row is: \[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix} = 8 \begin{vmatrix} 3 & 5 \\ 4 & 3 \end{vmatrix} - 2 \begin{vmatrix} 12 & 5 \\ 16 & 3 \end{vmatrix} + 7 \begin{vmatrix} 12 & 3 \\ 16 & 4 \end{vmatrix}. \] Step 3: Compute the \( 2 \times 2 \) minors.
1. For \( \begin{vmatrix} 3 & 5 \\ 4 & 3 \end{vmatrix} \): \[ \begin{vmatrix} 3 & 5 \\ 4 & 3 \end{vmatrix} = (3)(3) - (5)(4) = 9 - 20 = -11. \] 2. For \( \begin{vmatrix} 12 & 5 \\ 16 & 3 \end{vmatrix} \): \[ \begin{vmatrix} 12 & 5 \\ 16 & 3 \end{vmatrix} = (12)(3) - (5)(16) = 36 - 80 = -44. \] 3. For \( \begin{vmatrix} 12 & 3 \\ 16 & 4 \end{vmatrix} \): \[ \begin{vmatrix} 12 & 3 \\ 16 & 4 \end{vmatrix} = (12)(4) - (3)(16) = 48 - 48 = 0. \] Step 4: Substitute the minor values back into the determinant expression.
\[ \begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix} = 8(-11) - 2(-44) + 7(0) = -88 + 88 + 0 = 0. \] Conclusion: The determinant evaluates to \( 0 \).