Consider a homogeneous system of three linear | vedclass

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

Question

(B) For a homogeneous system $AX=O$, if the system has non-trivial solutions (infinite solutions), then the determinant of the matrix $A$ must be zero

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) For a homogeneous system $AX=O$, if the system has non-trivial solutions (infinite solutions), then the determinant of the matrix $A$ must be zero, i.e.,$|A| = 0$. This implies that the rank of $A$ must be less than the number of variables, which is $3$. Given that $X = [l, m, 0]^T$ is a solution with $l, m 
eq 0$, the solution space contains at least one non-zero vector. Since the solution is of the form $k[l, m, 0]^T$, the dimension of the null space (nullity) is at least $1$. By the Rank-Nullity Theorem, $	ext{rank}(A) + 	ext{nullity}(A) = n$, where $n=3$. If the nullity is $1$, then $	ext{rank}(A) = 3 - 1 = 2$. If the nullity were $2$, the solution space would be a plane, but the given form suggests a line of solutions. Thus, the rank of $A$ is $2$.

Consider a homogeneous system of three linear equations in three unknowns represented by $AX=O$. If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right]$, where $l \neq 0, m \neq 0, l, m \in \mathbb{R}$, represents an infinite number of solutions of this system, then the rank of $A$ is:

Similar Questions

If ${D_p} = \begin{vmatrix} p & 15 & 8 \\ p^2 & 35 & 9 \\ p^3 & 25 & 10 \end{vmatrix}$,then ${D_1} + {D_2} + {D_3} + {D_4} + {D_5} = $

Suppose $\alpha, \beta, \gamma$ are the roots of the equation $x^3+qx+r=0$ (where $r \neq 0$) and they are in $A$.$P$. Then the rank of the matrix $\begin{bmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{bmatrix}$ is

In a matrix $A$,if all the sub-matrices of order $k$ are singular and there is at least one non-singular sub-matrix of order $r$ $(r < k)$,then the rank $(\rho)$ of the matrix $A$:

Let $f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right|$,then the maximum value of $f(x)$ is:

The rank of the matrix $A=\begin{bmatrix} 1 & -1 & 0 & -2 \\ -4 & 4 & 0 & 8 \\ -2 & 1 & 2 & 4 \end{bmatrix}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module