Which of the following determinant(s) vanish( | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For option $A$: Let $\Delta_1 = \left| \begin{array}{ccc} 1 & bc & bc(b+c) \ 1 & ca & ca(c+a) \ 1 & ab & ab(a+b) \end{array} ight|$. Multiply

Vedclass · Accepted Answer

$\left| \begin{array}{ccc} 1 & bc & bc(b+c) \ 1 & ca & ca(c+a) \ 1 & ab & ab(a+b) \end{array} ight|$

Vedclass · Answer

(A) For option $A$: Let $\Delta_1 = \left| \begin{array}{ccc} 1 & bc & bc(b+c) \ 1 & ca & ca(c+a) \ 1 & ab & ab(a+b) \end{array} ight|$. Multiply $C_1$ by $abc$,then $\Delta_1 = \frac{1}{abc} \left| \begin{array}{ccc} a & abc & abc(b+c) \ b & abc & abc(c+a) \ c & abc & abc(a+b) \end{array} ight|$. Taking $abc$ common from $C_2$ and $C_3$,$\Delta_1 = \frac{(abc)^2}{abc} \left| \begin{array}{ccc} a & 1 & b+c \ b & 1 & c+a \ c & 1 & a+b \end{array} ight| = abc \left| \begin{array}{ccc} a & 1 & a+b+c \ b & 1 & a+b+c \ c & 1 & a+b+c \end{array} ight|$. Since $C_1+C_2$ is proportional to $C_3$,$\Delta_1 = 0$.For option $B$: Let $\Delta_2 = \left| \begin{array}{ccc} 1 & ab & \frac{a+b}{ab} \ 1 & bc & \frac{b+c}{bc} \ 1 & ca & \frac{c+a}{ca} \end{array} ight|$. This determinant does not necessarily vanish for all $a, b, c$.For option $C$: Let $\Delta_3 = \left| \begin{array}{ccc} 0 & a-b & a-c \ b-a & 0 & b-c \ c-a & c-b & 0 \end{array} ight|$. This is a skew-symmetric determinant of odd order $3 	imes 3$. The value of a skew-symmetric determinant of odd order is always $0$. Thus,$\Delta_3 = 0$.

Which of the following determinant$(s)$ vanish(es)?

Similar Questions

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3I)(A - 5I) = O$,where $I = I_3$ and $O = O_3$. If $\alpha A + \beta A^{-1} = 4I$,then $\alpha + \beta$ is equal to

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then $A^3 - 4A^2 - 6A$ is equal to:

If the determinant $\Delta = \begin{vmatrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0$,then:

If $M$ and $N$ are square matrices of order $3$, then which one of the following statements is not true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module