Question:
Evaluate the determinant $\begin{vmatrix} 11 & 1 & 1 \\ 1 & 21 & 1 \\ 1 & 1 & 31 \end{vmatrix}$:
Evaluate the determinant $\begin{vmatrix} 11 & 1 & 1 \\ 1 & 21 & 1 \\ 1 & 1 & 31 \end{vmatrix}$:
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Use row operations to create zeros and simplify determinant calculation.
Updated On: Apr 24, 2026
- $7100$
- $6800$
- $7300$
- $6900$
- $6700$
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The Correct Option is A
Solution and Explanation
Concept:
• Use determinant expansion or simplification using row/column operations
Step 1: Apply row operation
Let $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1$ \[ = \begin{vmatrix} 11 & 1 & 1 -10 & 20 & 0 -10 & 0 & 30 \end{vmatrix} \]
Step 2: Expand determinant
Using first row: \[ = 11 \begin{vmatrix} 20 & 0 0 & 30 \end{vmatrix} - 1 \begin{vmatrix} -10 & 0 -10 & 30 \end{vmatrix} + 1 \begin{vmatrix} -10 & 20 -10 & 0 \end{vmatrix} \]
Step 3: Calculate minors
\[ = 11(20 \cdot 30) - ( -10 \cdot 30 ) + ( (-10 \cdot 0) - (20 \cdot -10) ) \] \[ = 11(600) - (-300) + (0 + 200) \] \[ = 6600 + 300 + 200 = 7100 \] Final Conclusion:
\[ = 7100 \]
• Use determinant expansion or simplification using row/column operations
Step 1: Apply row operation
Let $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1$ \[ = \begin{vmatrix} 11 & 1 & 1 -10 & 20 & 0 -10 & 0 & 30 \end{vmatrix} \]
Step 2: Expand determinant
Using first row: \[ = 11 \begin{vmatrix} 20 & 0 0 & 30 \end{vmatrix} - 1 \begin{vmatrix} -10 & 0 -10 & 30 \end{vmatrix} + 1 \begin{vmatrix} -10 & 20 -10 & 0 \end{vmatrix} \]
Step 3: Calculate minors
\[ = 11(20 \cdot 30) - ( -10 \cdot 30 ) + ( (-10 \cdot 0) - (20 \cdot -10) ) \] \[ = 11(600) - (-300) + (0 + 200) \] \[ = 6600 + 300 + 200 = 7100 \] Final Conclusion:
\[ = 7100 \]
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