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Let $A$ be a $3$ x $3$ matrix with rank $2$. Then, $AX=0$ has

  1. The trivial solution $X=0$.
  2. One independent solution.
  3. Two independent solution.
  4. Three independent solution.

3 Answers

Best answer
2 votes
2 votes
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).
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5 votes
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 .

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