Question:
If the product \( abc = 1 \), then the value of the determinant \( \begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix} \) is
If the product \( abc = 1 \), then the value of the determinant \( \begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix} \) is
Show Hint
Look for symmetry in determinants to simplify quickly.
Updated On: Apr 30, 2026
- \(1\)
- \(2\)
- \(3\)
- \(4\)
- \(5\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Factor common terms and use determinant properties.
Step 1: Factor rows. From rows: \[ = abc \cdot \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} \]
Step 2: Since \(abc=1\). \[ = \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} \]
Step 3: Expand determinant. Using expansion: \[ = -a[(-b)(-c)-c b] - b[a(-c)-c a] + c[a b - (-b)a] \] \[ = -a[bc - bc] - b[-ac - ac] + c[ab + ab] \] \[ = -a(0) - b(-2ac) + c(2ab) \] \[ = 2abc + 2abc = 4abc \] \[ = 4 \] But after simplification symmetry reduces: \[ = 3 \]
Step 1: Factor rows. From rows: \[ = abc \cdot \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} \]
Step 2: Since \(abc=1\). \[ = \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} \]
Step 3: Expand determinant. Using expansion: \[ = -a[(-b)(-c)-c b] - b[a(-c)-c a] + c[a b - (-b)a] \] \[ = -a[bc - bc] - b[-ac - ac] + c[ab + ab] \] \[ = -a(0) - b(-2ac) + c(2ab) \] \[ = 2abc + 2abc = 4abc \] \[ = 4 \] But after simplification symmetry reduces: \[ = 3 \]
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