GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 7

Answer 1

• A. False. E.g.: Suppose $\text{A}$ had $12$ identical nonzero rows (and the rest all zeroes) to start with, and we swapped two rows to get $\text{B}$, which also has exactly $12$ nonzero rows. The rank of $\text{A}$ is $1.$
• B. False. Follows from the definition; "row or column operations do not change the rank".
• C. True. Same reason as above; the null space is unchanged.
• D. True. $\text{B}$ has $11$ columns so its rank cannot be more.

Answer 2

Option A). It is clearly false

since Rank of the Matrix = at most min(number of column,number of rows).

so it will be at most 11.

Option B). It is kind of a hypocritical statement.

To find rank we are doing Gaussian elimination to convert matrix into row echelon form or simply we are doing row operations only.

so it simply doesn’t affect the rank.

Option C). True Rank isn’t affected by row operation, the number of pivots and number of free variables will remain the same.

Option D).  True already mentioned in option A.