your_file.csv

exam name,year,problem number,problem,solution,correct option
IIT JAM MS,2023,Problem 2,"Let $M$ = $M_1M_2$ , where $M_1$ and $M_2$ are two 3 × 3  distinct matrices. Consider the following two statements:
(I) The rows of $M$ are linear combinations of rows of $M_2$.
(II) The columns of $M$ are linear combinations of columns of $M_1$.

Then,

(A) only  (I) is TRUE

(B) only (II) is TRUE

(C) both (I) and (II) are TRUE

(D) neither (I) nor (II) is TRUE","Let $M_1$= $\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$  and  $M_2$ = $\begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}$

We can represent the matrix $M_1$ using it’s column vectors and $M_2$ using it’s row vectors as shown:

Let $M_1$ = $\begin{bmatrix} \vec{a_{1}}& \vec{a_{2}} &\vec{a_{3}}\\ \end{bmatrix}$ and $M_2$ = $\begin{bmatrix} \vec{b_{1}}\\ \vec{b_{2}}\\ \vec{b_{3}}\\ \end{bmatrix}$, where $\vec{a_i}$ and $\vec{b_i}$ is in  $R^3$

The matrix multiplication $M_1M_2$  yields a 3x3 matrix $M$. 

This matrix can be written in two different ways.

### Column View of $M$

$M$ = $\begin{bmatrix} \sum_{i=1}^{3} b_{i1}\vec{a_{i}} & \sum_{i=1}^{3} b_{i2}\vec{a_{i}} & \sum_{i=1}^{3} b_{i3}\vec{a_{i}}\\ \end{bmatrix}$. This shows that the each column of $M$  is a linear combinations of the columns of $M_1$.

### Row View of $M$

$M$ = $\begin{bmatrix} \sum_{i=1}^{3} a_{1i}\vec{b_{i}} \\ \sum_{i=1}^{3} b_{2i}\vec{b_{i}} \\ \sum_{i=1}^{3} b_{3i}\vec{b_{i}}\\ \end{bmatrix}$. 

This shows that the each row of $M$  is a linear combination of the rows of $M_2$.

You can algebraically verify this using the below matrix multiplication formula.

![Untitled](https://prod-files-secure.s3.us-west-2.amazonaws.com/aafb3d2f-7f27-4753-a17f-8ceef5abd2ad/a8943351-cd19-44c7-a996-c6438c21eb3f/Untitled.png)

However, you can extend the proof similarly to show that this is true for any $n \times n$ matrix.",C