The determinant left| begin{array}{ccc} a & b | vedclass

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

Question

(B) Let $\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ a\alpha + b & b\alpha + c & 0 \end{array} ight|$.Applying

Vedclass · Accepted Answer

$G. P.$

Vedclass · Answer

(B) Let $\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ a\alpha + b & b\alpha + c & 0 \end{array} ight|$.Applying the row operation $R_3 	o R_3 - \alpha R_1 - R_2$:$\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ 0 & 0 & -(a\alpha^2 + 2b\alpha + c) \end{array} ight|$.Expanding along the third row $(R_3)$:$\Delta = -(a\alpha^2 + 2b\alpha + c) \cdot \left| \begin{array}{cc} a & b \ b & c \end{array} ight| = -(a\alpha^2 + 2b\alpha + c)(ac - b^2) = (b^2 - ac)(a\alpha^2 + 2b\alpha + c)$.For $\Delta = 0$,we must have $b^2 - ac = 0$ or $a\alpha^2 + 2b\alpha + c = 0$.The condition $b^2 - ac = 0$ implies $b^2 = ac$,which means $a, b, c$ are in $G.P.$

The determinant $\left| \begin{array}{ccc} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{array} \right| = 0$,if $a, b, c$ are in

Similar Questions

If $A = \begin{bmatrix} 5 & 5x & x \\ 0 & x & 5x \\ 0 & 0 & 5 \end{bmatrix}$ and $|A^2| = 25$,then $|x|$ is equal to

The value of $\det A$,where $A = \begin{bmatrix} 1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$,lies

If the determinant of the matrix $A = \begin{bmatrix} 0 & a & b \\ -a & 0 & \beta \\ -b & -\alpha & 0 \end{bmatrix}$ is zero for all $a, b$,then $\alpha + \beta =$

The value of $\left|\begin{array}{lll}\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ \sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ \tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$ is

If $2x + 3y - 5z = 7$,$x + y + z = 6$,and $3x - 4y + 2z = 1$,then $x =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module