If the determinant of the matrix A = begin{bm | vedclass

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

Question

(A) The determinant of the matrix $A$ is given by $|A| = \begin{vmatrix} 0 & a & b \ -a & 0 & \beta \ -b & -\alpha & 0 \end{vmatrix} = 0$. Expanding

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) The determinant of the matrix $A$ is given by $|A| = \begin{vmatrix} 0 & a & b \ -a & 0 & \beta \ -b & -\alpha & 0 \end{vmatrix} = 0$. Expanding the determinant along the first row: $|A| = 0(0 - (\beta)(-\alpha)) - a(0 - (\beta)(-b)) + b((-a)(-\alpha) - 0) = 0$. $|A| = 0 - a(b\beta) + b(a\alpha) = 0$. $-ab\beta + ab\alpha = 0$. $ab(\alpha - \beta) = 0$. Since this holds for all $a, b$, we must have $\alpha - \beta = 0$, which implies $\alpha = \beta$. However, checking the original matrix structure provided in the prompt: $\begin{bmatrix} 0 & a & b \ -a & 0 & \beta \ -b & \alpha & 0 \end{bmatrix}$. Expanding this: $0(0 - \alpha\beta) - a(0 - (\beta)(-b)) + b((-a)(\alpha) - 0) = 0$. $-ab\beta - ab\alpha = 0$. $-ab(\alpha + \beta) = 0$. Since this holds for all $a, b$, we have $\alpha + \beta = 0$.

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 $\alpha + \beta =$

Similar Questions

If $\left|\begin{array}{ccc}\cos (A+B) & -\sin (A+B) & \cos 2 B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{array}\right|=0$, then $B$ is equal to

The area of the triangle whose vertices are $(3,5), (2,2)$ and $(k, 2)$ is $3$ sq. unit. Then,the value of $k$ is . . . . . . .

If $x$ is a positive integer,then $\Delta = \left| \begin{array}{ccc} x! & (x+1)! & (x+2)! \\ (x+1)! & (x+2)! & (x+3)! \\ (x+2)! & (x+3)! & (x+4)! \end{array} \right|$ is equal to

The area of $\triangle PQR$ with the vertices $P(k, 1)$,$Q(2, 4)$,and $R(1, 1)$ is $3$ sq. units. Then,$k = $ . . . . . . .

If $\left[\begin{array}{ccc}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{array}\right]$ has no inverse,then the real value of $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module