A, C are 3 times 3 matrices. B, D are 3 times | vedclass

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

Question

(D) For the system $AX=B$, since it has a unique solution, the matrix $A$ must be invertible, which implies $\operatorname{rank}(A) = 3$. Thus, the au

Vedclass · Accepted Answer

$\operatorname{rank}([A: D]) \geq \operatorname{rank}([C: B])$

Vedclass · Answer

(D) For the system $AX=B$, since it has a unique solution, the matrix $A$ must be invertible, which implies $\operatorname{rank}(A) = 3$. Thus, the augmented matrix $[A: B]$ also has rank $3$.For the system $CX=D$, since it has an infinite number of solutions, the matrix $C$ must be singular, which implies $\operatorname{rank}(C) Comparing the ranks: $\operatorname{rank}(A) = 3$ and $\operatorname{rank}(C)  \operatorname{rank}(C)$.Regarding the augmented matrices, $\operatorname{rank}([A: D])$ is at most $3$, and $\operatorname{rank}([C: B])$ is at most $3$. Since $\operatorname{rank}(A) = 3$, the rank of $[A: D]$ is $3$. Since $\operatorname{rank}(C) < 3$, the rank of $[C: B]$ is at most $3$. Thus, $\operatorname{rank}([A: D]) \geq \operatorname{rank}([C: B])$ is a true statement.

$A, C$ are $3 \times 3$ matrices. $B, D$ are $3 \times 1$ matrices. If $AX=B$ has a unique solution and $CX=D$ has an infinite number of solutions, then:

Similar Questions

Let the system of linear equations $x+y+kz=2$; $2x+3y-z=1$; $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$; $(2k+1)x+(k+5)y=10$ has:

If the system of linear equations,$x+y+z = 6$,$x+2y+3z = 10$,and $3x+2y+\lambda z = \mu$ has more than two solutions,then $\mu-\lambda^{2}$ is equal to

Given the system of linear equations: $2x + 3y + 4z = 9$,$4x + 9y + 3z = 10$,and $5x + 10y + 5z = 11$. The value of $x$ is given by:

If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions,then $\alpha + \beta - \alpha \beta$ is equal to .... .

If the system of linear equations $x - 4y + 7z = g$,$3y - 5z = h$,and $-2x + 5y - 9z = k$ is consistent,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module