If {a_1}x + {b_1}y + {c_1}z = 0, {a_2}x + {b_ | vedclass

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

Question

(D) The given system of equations is a homogeneous system of linear equations of the form $AX = 0$,where $A$ is the coefficient matrix and $X = [x, y,

Vedclass · Accepted Answer

Infinite solution

Vedclass · Answer

(D) The given system of equations is a homogeneous system of linear equations of the form $AX = 0$,where $A$ is the coefficient matrix and $X = [x, y, z]^T$.For a homogeneous system $AX = 0$,the nature of the solution depends on the determinant of the coefficient matrix,denoted by $|A|$.If $|A| 
eq 0$,the system has only the trivial solution $(x = 0, y = 0, z = 0)$.If $|A| = 0$,the system has infinitely many non-trivial solutions in addition to the trivial solution.Since the problem states that $\left| \begin{matrix} {a_1} & {b_1} & {c_1} \ {a_2} & {b_2} & {c_2} \ {a_3} & {b_3} & {c_3} \end{matrix} ight| = 0$,the determinant of the coefficient matrix is zero.Therefore,the system has infinitely many solutions.

If ${a_1}x + {b_1}y + {c_1}z = 0, {a_2}x + {b_2}y + {c_2}z = 0, {a_3}x + {b_3}y + {c_3}z = 0$ and $\left| \begin{matrix} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{matrix} \right| = 0$,then the given system has

Similar Questions

For what values of $a$ will the system of equations $x+y+z=1$, $2x+3y+2z=2$, and $ax+ay+2az=4$ have a unique solution?

Let $S$ be the set of values of $\lambda$,for which the system of equations
$6 \lambda x - 3 y + 3 z = 4 \lambda^2$
$2 x + 6 \lambda y + 4 z = 1$
$3 x + 2 y + 3 \lambda z = \lambda$
has no solution. Then $12 \sum_{\lambda \in S} |\lambda|$ is equal to $...........$.

If $3X + 2Y = I$ and $2X - Y = O$,where $I$ and $O$ are unit and null matrices of order $3$ respectively,then

If the system of equations $\alpha x + y + z = 5$,$x + 2y + 3z = 4$,and $x + 3y + 5z = \beta$ has infinitely many solutions,then the ordered pair $(\alpha, \beta)$ is equal to:

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module