GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 21

Answer 1

option ; a , b, c, d are correct one . 

Comments

but M = No. Of rows. In the previous image too,when n=1 , m was 3 only....

so what was the need for the 2nd image and taking n = 2 ( no. of columns as 2 ) ??

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To check rank(A)=2 possible.

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Further adding to this answer by @Gutley>>  n=1 or 2 only, it can't be 3 or >3 as n=3 mein for B1 no slotuion case would not be possible and n>3 mein unique solution would not be possible for B2

Answer 2

Since the question is Which of them CAN be true, it means that we have to just find one example to show that the option is possible.
 

Option	No Solution	Unique Solution
A & B	$$\begin{bmatrix} a & b & 1 \\ 0 & c & 1 \\ 0 & 0 & 1 \end{bmatrix}$$	$$\begin{bmatrix} a & b & 0 \\ 0 & c & 1 \\ 0 & 0 & 0 \end{bmatrix}$$
C & D	$$\begin{bmatrix} a & 1 \\ 0 & 1 \\ 0 & 1 \end{bmatrix}$$	interchanging R1 and R2 $$\begin{bmatrix} a & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$$

,where all the alphabets are non-zero numbers.

Answer 3

Since b is a 3x1 matrix obtained from Ax, we can easily infer that the dimensions of A and x must be 3xn and nx1. So, if A is a mxn matrix, where m must be equal to 3.

If we have a unique solution for b and, at the same time, no solution for another b, it means that A has linearly independent column vectors, but they are not covering the entire space. So, the number of linearly independent vectors in matrix A must be less than the dimension of b, which is less than 3. So, n should be either 2 or 1. So, the rank can also be either 2 or 1.