GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 2

Answer 1

Number of Columns in A = 5

Number of free varibles = 2 ( c and e represent the free variable )  

Rank = Number of Columns – No. of free Variables (Linear Dependent Columns  ) .

Rank = 5 – 2  = 3

 

 

Answer is 3 .

Comments

Please correct the statement.

Wrong statement :- Rank = Number of Columns – No. of free Variables (Linear Dependent Rows)
It should be linearly dependent columns (not Linear Dependent Rows) because variables represent columns and not rows.

Answer 2

We can re-write the given solution in terms of ‘c’ and ‘e’ as:

 $\begin{bmatrix} \\ 6c-12e \\ -4c+10e \\ c \\ -5e \\e \end{bmatrix}$ = c $\begin{bmatrix} \\ 6 \\ -4 \\ 1 \\ 0 \\0 \end{bmatrix}$ + e$\begin{bmatrix} \\ -12 \\ 10 \\ 0 \\ -5 \\1\end{bmatrix}$.

It means we have 2 free columns.

So, no. of pivot columns = Total no. of columns – No. of free columns = 5 – 2 = 3

Rank = No. of pivot columns = 3.

Comments

well formula is correct but in this question there a bit different term need to use
its,
Rank = dim(Basis) - no.of span variables.
Rank Nullity Theorem can be expressed in Different forms do use wisely.

Answer 3

Since, A is a 4x5 Matrix
Therefore, it means that A has 4 Linear Equations with 5 variables.

Now, you can write the x in Ax = 0 as:

x =

This means that x has 2 Linear Independent variables.
This will only happen when A has 2 Free Variables.

It's a fact that,
Total number of Variables = Number of Pivot Variables + Number of Free Variables
5 = Number of Pivot Variables + 2
Number of Pivot Variables (or Rank of A) = 3
 

Answer 4

1. The solution given is a linear combination of 2 linearly independent vectors.
2. So, we have 2 linearly independent solutions. We know that the nullity of a matrix is no.of Linearly independent solutions for a matrix

Therefore the nullity of given matrix is 2.

hence, the rank is 5 - 2 = 3