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DebRC
asked
in Linear Algebra
Sep 19, 2022

383 views
2 votes

Best answer

The answer will be option A,B.

Since rank of matrix A is 2, so |A| = 0.This says that there is a non-trivial and non-zero solution. Non-trivial solution means along with a zero solution(0,0,0) there are many points of intersection in the plane.

Number of independent solutions or number of free variables = Total number of variables – rank of A

= 3 – 2 = 1

For eg:

x1 + x2 + x3 = 0

[1 1 1]$\begin{bmatrix} x1\\ x2\\ x3 \end{bmatrix}$ = [0]

Rank = 1 and total variables are 3.

Let x2 =k, x3=t , then x1= – k – t

Total independent variables are 2 (x2,x3) and dependent variable is 1 (x1).

Since rank of matrix A is 2, so |A| = 0.This says that there is a non-trivial and non-zero solution. Non-trivial solution means along with a zero solution(0,0,0) there are many points of intersection in the plane.

Number of independent solutions or number of free variables = Total number of variables – rank of A

= 3 – 2 = 1

For eg:

x1 + x2 + x3 = 0

[1 1 1]$\begin{bmatrix} x1\\ x2\\ x3 \end{bmatrix}$ = [0]

Rank = 1 and total variables are 3.

Let x2 =k, x3=t , then x1= – k – t

Total independent variables are 2 (x2,x3) and dependent variable is 1 (x1).

@samarpita sorry my bad .. i have wrongly interpret the comment as there is only trivial solution possible .

0

5 votes

A is a $3*3$ matrix .

Rank of A is $2$ which implies it has $2$ linearly independent vectors .

Now $AX=0$ has only two cases either it will have unique solution / trivial solution which is all the variable will contain 0 as solution .

Or it will have infinitely many solutions .

In case of $AX=0$ no solution case is not possible as (0,0,0) always will be solution for this kind of system of equation .

Now , $AX=0$ will have trivial solution only when $\left | A \right |\neq 0$ .

$AX=0$ will have infinitely many solution when $\left | A \right |=0$ . Now As in the problem $A$ is $3*3$ matrix and rank of $A$ =2 which implies the determinant of $A$ is equal to $0$ .

So this conclude the system here $AX=0$ has infinitely many solutions .

Rank of A is $2$ which implies it has $2$ linearly independent vectors .

Now $AX=0$ has only two cases either it will have unique solution / trivial solution which is all the variable will contain 0 as solution .

Or it will have infinitely many solutions .

In case of $AX=0$ no solution case is not possible as (0,0,0) always will be solution for this kind of system of equation .

Now , $AX=0$ will have trivial solution only when $\left | A \right |\neq 0$ .

$AX=0$ will have infinitely many solution when $\left | A \right |=0$ . Now As in the problem $A$ is $3*3$ matrix and rank of $A$ =2 which implies the determinant of $A$ is equal to $0$ .

So this conclude the system here $AX=0$ has infinitely many solutions .

edited
Sep 20, 2022
by DebRC

Answer given by them was (a) and (b).

Reason they stated –

If $r$ is the rank of matrix A and $n * n$ is the order of matrix then we shall have $(n– r)$ linearly independent non-trivial solutions. Any linear combination of these $(n– r)$ solutions will also be a solution of $AX = 0$.

I am not being able to understand the second sentence of their statement.

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