Question

The value of the determinant \[ \begin{vmatrix} 50 & 15 & 3\\ 51 & 17 & 7\\ 27 & 9 & 6 \end{vmatrix} \] is:

• A. $-195$
• B. $185$
• C. $190$
• D. $195$

Hint:

Expand along the row/column with simplest values to reduce calculations.

Answer 1

The Correct Option is D

Concept: Expand determinant along first row.
Step 1: Apply formula.
\[ = 50 \begin{vmatrix} 17 & 7\\ 9 & 6 \end{vmatrix} - 15 \begin{vmatrix} 51 & 7\\ 27 & 6 \end{vmatrix} + 3 \begin{vmatrix} 51 & 17\\ 27 & 9 \end{vmatrix} \]
Step 2: Calculate minors.
\[ \begin{vmatrix} 17 & 7\\ 9 & 6 \end{vmatrix} = 102 - 63 = 39 \] \[ \begin{vmatrix} 51 & 7\\ 27 & 6 \end{vmatrix} = 306 - 189 = 117 \] \[ \begin{vmatrix} 51 & 17\\ 27 & 9 \end{vmatrix} = 459 - 459 = 0 \]
Step 3: Substitute.
\[ = 50(39) - 15(117) + 3(0) \] \[ = 1950 - 1755 = 195 \]
Hence, the value is 195.