Question:
If \( \Delta = \begin{vmatrix}1& 2& 3 \\ 2& 3& 5 \\ 3& 6& 12\end{vmatrix} \) and
\( \Delta' = \begin{vmatrix}4& 8& 15 \\ 3& 6& 12 \\ 2& 3& 5\end{vmatrix} \), then
If \( \Delta = \begin{vmatrix}1& 2& 3 \\ 2& 3& 5 \\ 3& 6& 12\end{vmatrix} \) and
\( \Delta' = \begin{vmatrix}4& 8& 15 \\ 3& 6& 12 \\ 2& 3& 5\end{vmatrix} \), then
Show Hint
Track row operations carefully: swaps change sign, scaling multiplies determinant.
Updated On: May 8, 2026
- \( \Delta' = 2\Delta \)
- \( \Delta' = -2\Delta \)
- \( \Delta' = \Delta \)
- \( \Delta' = -\Delta \)
- \( \Delta' = 3\Delta \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Use determinant properties:
• Row interchange changes sign
• Multiplying row multiplies determinant
Step 1: Compare matrices
Observe \( \Delta' \) rows are rearranged and scaled.
Step 2: Reorder rows
Swap rows to match structure → introduces sign change.
Step 3: Identify scaling
First row is roughly 2 times original row.
Step 4: Combine effects
Scaling factor = 2 Row swap factor = -1 \[ \Delta' = -2\Delta \]
Step 5: Final answer
\[ \boxed{-2\Delta} \]
• Row interchange changes sign
• Multiplying row multiplies determinant
Step 1: Compare matrices
Observe \( \Delta' \) rows are rearranged and scaled.
Step 2: Reorder rows
Swap rows to match structure → introduces sign change.
Step 3: Identify scaling
First row is roughly 2 times original row.
Step 4: Combine effects
Scaling factor = 2 Row swap factor = -1 \[ \Delta' = -2\Delta \]
Step 5: Final answer
\[ \boxed{-2\Delta} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Properties of Determinants Questions
- The value of the determinant $\begin{vmatrix}\sin ^{2} 36^{\circ} & \cos ^{2} 36^{\circ} & \cot 135^{\circ} \\ \sin ^{2} 53^{\circ} & \cot 135^{\circ} & \cos ^{2} 53^{\circ} \\ \cot 135^{\circ} & \cos ^{2} 25^{\circ} & \cos ^{2} 65^{\circ}\end{vmatrix}$ is
Let A be an invertible matrix of size 4x4 with complex entries. If the determinant of adj(A) is 5,then the number of possible value of determinant of A is
- Let \( A \) be a \( 3 \times 3 \) matrix with \( |A| = 7 \). If \( B = 3A \), then the value of \[ \left| \frac{{adj } A}{B} \right| \] is equal to
- If \[ A = \begin{bmatrix} -7 & 3 \\ 3 & -1 \end{bmatrix} \] then \( \det(A^5) \) is equal to:
- Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions