Question:
If \( \Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} \) and \( \Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix} \), then
If \( \Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} \) and \( \Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix} \), then
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Tip 1: For diagonal or triangular matrices, don't waste time expanding; just multiply the diagonal.
Tip 2: In \( \Delta_2 \), observe it's almost \( \Delta_1 \) with \( R_1, R_2 \) swapped and one element doubled.
Tip 2: In \( \Delta_2 \), observe it's almost \( \Delta_1 \) with \( R_1, R_2 \) swapped and one element doubled.
Updated On: Sep 10, 2026
- \( \Delta_1 = 2\Delta_2 \)
- \( \Delta_2 = -2\Delta_1 \)
- \( \Delta_1 = \Delta_2 \)
- \( \Delta_2 = -\Delta_1 \)
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The Correct Option is B
Solution and Explanation
Concept:
• The determinant of a diagonal matrix is the product of its diagonal elements.
• Interchanging any two rows (or columns) of a determinant changes its sign.
• If a scalar is multiplied to a row, the determinant value is multiplied by that scalar.
Step 1: Calculate the value of \( \Delta_1 \)
\( \Delta_1 \) is a diagonal matrix: \[ \Delta_1 = 1 \times 2 \times 3 = 6 \]
Step 2: Calculate the value of \( \Delta_2 \)
Expand \( \Delta_2 \) along the first row: \[ \Delta_2 = 0 - 2(1 \times 6 - 0) + 0 \] \[ \Delta_2 = -2(6) = -12 \]
Step 3: Relate the two values
We have \( \Delta_1 = 6 \) and \( \Delta_2 = -12 \). To find the relation, divide \( \Delta_2 \) by \( \Delta_1 \): \[ \frac{\Delta_2}{\Delta_1} = \frac{-12}{6} = -2 \] \[ \Delta_2 = -2\Delta_1 \]
• The determinant of a diagonal matrix is the product of its diagonal elements.
• Interchanging any two rows (or columns) of a determinant changes its sign.
• If a scalar is multiplied to a row, the determinant value is multiplied by that scalar.
Step 1: Calculate the value of \( \Delta_1 \)
\( \Delta_1 \) is a diagonal matrix: \[ \Delta_1 = 1 \times 2 \times 3 = 6 \]
Step 2: Calculate the value of \( \Delta_2 \)
Expand \( \Delta_2 \) along the first row: \[ \Delta_2 = 0 - 2(1 \times 6 - 0) + 0 \] \[ \Delta_2 = -2(6) = -12 \]
Step 3: Relate the two values
We have \( \Delta_1 = 6 \) and \( \Delta_2 = -12 \). To find the relation, divide \( \Delta_2 \) by \( \Delta_1 \): \[ \frac{\Delta_2}{\Delta_1} = \frac{-12}{6} = -2 \] \[ \Delta_2 = -2\Delta_1 \]
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