GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 8

Answer 1

option b is correct one 

Comments

In option d , i think you explained about consistency. statement is “ If a system of 25 linear equation in 13 unknown has atleast one solution then it has infinitely many solution“ means solution exists and we need to say about whether infinite or not, we can’t discuss about no solution in optiod D. There is a possibility that we have 13 pivots i.e, 13 LI vectors. that means we will have unique solution in that case. Which makes Option D wrong.  

@Sachin Mittal 1 kindly verify my assessment of Option D.

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I think yes you are correct.

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one solution is possible but to have infinite solutin we need to fill the space to make it LD. which is not possible here because R^25 and column is only 13

Answer 2

Option A: Lets have a echelon matrix A which 2x3 (n > m)

$$\begin{bmatrix} a & b & c\\ 0 & e & f \end{bmatrix}$$
In this example, c1 and c2 are pivot variables. We can see that c3 is a free variable,
that is, when n>m, not all the columns can become pivot variables.

Therefore, A is False.

Option B: Consistent means that the equation has atleast one solution for all b.
In other words, it means that the solutions exists for all b.

Now, the condition for No solution is that for an Augmented Matrix [ A | b ]
If any row of the matrix looks like this:
$$\begin{bmatrix} 0 & 0 & x \end{bmatrix}$$
where x != 0

But in the options, it has been given that every row has a pivot. In other words, every row has a non-zero variable.
Therefore, the condition for No solutions fails and the equation is consistent. 

B is True.

Option C: The matrix it is talking about is A which is 25 x 13, that is m = 25 and n = 13.

Since, it has only 13 variables, that means only 13 pivots are possible.
25-13 = 12 rows have to be 0 in echelon form.
but it's not given that b in (Ax = b), will be zero or not.
So, whether the solution exists or not, will depend on b for that row.

If b != 0, then it will satisfy the No solution condition otherwise it will not.

Option C is False.

Option D: For the same matrix in Option C. Now, it has been given that atleast one solutions exists for all b.
So, b = 0 for the rows which are 0 in echelon form. 

For the solution to be infinite, there has to be a free variable in row echelon form, but since the matrix has 25 rows and only 13 columns. Therefore, all the 13 columns (or variables) will become pivot and no free variables will be there. 
Therefore, the solution will be unique if it exists. 

D is False

 

Answer 3

@Sachin Mittal 1 sir for option A even though pivot in every column there we can still have more rows right

for eg

1 2 1

0 1 0

0 0 4

0 0 0

sir here m>n and every col have pivots .

Please correct me if i am wrong

Comments

Yes thats right.
Answer is B only.
Can you please convert your answer to comment or make it a full answer.

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yes but yesterday ans given was A also that’s why I asked but now edited

Thank u sir