The positive value of a for which the system | vedclass

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

Question

(C) For a system of linear homogeneous equations to have non-trivial solutions,the determinant of the coefficient matrix must be equal to $0$.The coef

Vedclass · Accepted Answer

$\frac{1+\sqrt{5}}{2}$

Vedclass · Answer

(C) For a system of linear homogeneous equations to have non-trivial solutions,the determinant of the coefficient matrix must be equal to $0$.The coefficient matrix is $A = \begin{bmatrix} 1 & a & 1 \ a & 2 & -1 \ 2 & 3 & 1 \end{bmatrix}$.Setting the determinant $|A| = 0$:$1(2(1) - (-1)(3)) - a(a(1) - (-1)(2)) + 1(a(3) - 2(2)) = 0$$1(2 + 3) - a(a + 2) + 1(3a - 4) = 0$$5 - a^2 - 2a + 3a - 4 = 0$$-a^2 + a + 1 = 0$$a^2 - a - 1 = 0$Using the quadratic formula $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:$a = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}$.Since we are looking for the positive value of $a$,we take $a = \frac{1+\sqrt{5}}{2}$.

The positive value of $a$ for which the system of linear homogeneous equations $x+ay+z=0$,$ax+2y-z=0$,and $2x+3y+z=0$ has non-trivial solutions is

Similar Questions

If $x = \alpha, y = \beta, z = \gamma$ is the unique solution of the system of equations $5x - 2y + 3z = 0$,$7x + 10y - 8z = 3$ and $2x + 3y - 4z = -4$,then $\beta =$

If the system of equations $x+2y+3z=3$,$4x+3y-4z=4$,and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions,then the ordered pair $(\lambda, \mu)$ is equal to

Consider the system of equations: $x + ay = 0$,$y + az = 0$,and $z + ax = 0$. The set of all real values of $a$ for which the system has a unique solution is:

$A$ and $C$ lie in $\left[0, \frac{\pi}{2}\right)$ and $B$ lies in $[0, 2\pi]$. If $\tan A + 3 \cos B + 6 \sin C = 1$; $3 \tan A + \cos B + 4 \sin C = 4$; $5 \tan A + 3 \cos B - 8 \sin C = -2$,then $B - 2A - C =$

If the system of equations
$2x + 7y + \lambda z = 3$
$3x + 2y + 5z = 4$
$x + \mu y + 32z = -1$
has infinitely many solutions,then $(\lambda - \mu)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module