Question

The value of \[ \Delta = \begin{vmatrix} 42 & 2 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{vmatrix} \] is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( -3 \)
• D. \( -15 \)

Hint:

For determinants of \( 3 \times 3 \) matrices, expand along any row or column using minors and cofactors.

Answer 1

The Correct Option is A

The determinant is calculated as: \[ \Delta = 42 \begin{vmatrix} 7 & 9 \\ 5 & 3 \end{vmatrix} - 2 \begin{vmatrix} 79 & 9 \\ 29 & 3 \end{vmatrix} + 5 \begin{vmatrix} 79 & 7 \\ 29 & 5 \end{vmatrix}. \] Computing the minors: \[ \begin{vmatrix} 7 & 9 \\ 5 & 3 \end{vmatrix} = (7 \times 3) - (9 \times 5) = 21 - 45 = -24. \] \[ \begin{vmatrix} 79 & 9 \\ 29 & 3 \end{vmatrix} = (79 \times 3) - (9 \times 29) = 237 - 261 = -24. \] \[ \begin{vmatrix} 79 & 7 \\ 29 & 5 \end{vmatrix} = (79 \times 5) - (7 \times 29) = 395 - 203 = 192. \] Substituting back: \[ \Delta = (42 \times -24) - (2 \times -24) + (5 \times 192). \] \[ = -1008 + 48 + 960 = 0. \]