Question:
The value of determinant $\begin{vmatrix}a-b&b+c&a\\ b-a&c+a&b\\ c-a&a+b&c \end{vmatrix}$ is
The value of determinant $\begin{vmatrix}a-b&b+c&a\\ b-a&c+a&b\\ c-a&a+b&c \end{vmatrix}$ is
Updated On: Apr 17, 2024
- 1
- 0
- 2
- 3
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Operating $R_1\rightarrow R_1-R_2$
$R_2\rightarrow R_2-R_3$
$\begin{vmatrix}0 & a-b & (a-b)(a+b+c)\\0&b-c&(b-c)(a+b+c)\\1 &c&c^2-ab\end{vmatrix}$
$\Rightarrow (a-b)(b-c)\begin{vmatrix}0 &1 &a+b+c\\0 &1&a+b+c\\1&c&c^2-ab\end{vmatrix}=0$
Because $R_1$ and $R_2$ are identical.
Hence (b) is the correct answer.
$R_2\rightarrow R_2-R_3$
$\begin{vmatrix}0 & a-b & (a-b)(a+b+c)\\0&b-c&(b-c)(a+b+c)\\1 &c&c^2-ab\end{vmatrix}$
$\Rightarrow (a-b)(b-c)\begin{vmatrix}0 &1 &a+b+c\\0 &1&a+b+c\\1&c&c^2-ab\end{vmatrix}=0$
Because $R_1$ and $R_2$ are identical.
Hence (b) is the correct answer.
Was this answer helpful?
0
0
Top Questions on Properties of Determinants
- If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:
- JEE Main - 2024
- Mathematics
- Properties of Determinants
- If \[ A = \begin{bmatrix} -7 & 3 \\ 3 & -1 \end{bmatrix} \] then \( \det(A^5) \) is equal to:
- KEAM - 2024
- Mathematics
- Properties of Determinants
- 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
- KEAM - 2024
- Mathematics
- Properties of Determinants
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
- KEAM - 2023
- Mathematics
- Properties of Determinants
- If for some $c < 0$, the quadratic equation, $2cx^{2}-2\left(2c-1\right)x+3c^{2}=0$ has two distinct real roots, $\frac{1}{a}$ and $\frac{1}{b},$ then the value of the determinant. $\begin{vmatrix}1+a&1&1\\ 1&1+b&1\\ 1&1&1+c\end{vmatrix}$ is :
- JEE Main - 2020
- Mathematics
- Properties of Determinants
View More Questions
Questions Asked in KCET exam
The circuit shown in the figure contains two ideal diodes \( D_1 \) and \( D_2 \). If a cell of emf 3V and negligible internal resistance is connected as shown, then the current through \( 70 \, \Omega \) resistance (in amperes) is:
- KCET - 2025
- Refractive index
- The mean deviation about the mean for the data \( 4, 7, 8, 9, 10, 12, 13, 17 \) is:
- KCET - 2025
- measurement of angles
- The distance of the point \( P(-3,4,5) \) from the yz-plane is:
- KCET - 2025
- Distance of a Point From a Line
- If 'a' and 'b' are the order and degree respectively of the differentiable equation \[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + x^4 = 0, \quad \text{then} \, a - b = \, \_ \_ \]
- KCET - 2025
- Differential equations
- If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:
- KCET - 2025
- Binomial theorem
View More Questions
Concepts Used:
Properties of Determinants
Properties of Determinants:
- Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged.
- Switching Property: The interchanging of any two rows (or columns) of the Determinant changes its signs.
- All- Zero Property: The Determinants will be equivalent to zero if each term of rows and columns is zero.
- Proportionality (Repetition) Property: If all elements of a row (or column) are proportional or identical to the elements of some other row (or column), then the determinant is zero.
- Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
- Sum Property: If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
- Property of Invariance: If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + kCj
- Factor Property: If a Determinant Δ becomes 0 while considering the value of x = α, then (x - α) is considered as a factor of Δ
- Triangle Property: If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.
Read More: Properties of Determinants