GO Classes DPP | GATE CS, DA | Linear Algebra | Matrices

Answer 1

 

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Analysis of Matrix Operations

Statement A: General Column Operations

If $B$ is obtained from $A$ by general column operations, then $B = AM$ for some matrix $M$. However, general column operations are not necessarily reversible.

Counter-example:

Let

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

If we replace the second column with the zero column:

$$B = \begin{bmatrix} 1 & 0 \\ 3 & 0 \end{bmatrix}$$

The original information in the second column is lost. $A$ cannot be recovered from $B$.

Conclusion: Statement A is False.

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Statement B: Elementary Column Operations

Elementary column operations consist of:

1. Swapping two columns.

2. Multiplying a column by a non-zero constant $k$.

3. Adding a multiple of one column to another.

Each of these is inherently reversible.

Example:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Swap the columns to get $B$:

$$B = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}$$

Swapping them again returns $A$. Scaling by $k$ is reversed by scaling by $1/k$. Adding $nC_1$ to $C_2$ is reversed by subtracting $nC_1$ from $C_2$.

Conclusion: Statement B is True.

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Statement C: Elementary Row Operations

Elementary row operations follow the same logic as column operations but are applied to rows ($R_i$). They are:

1. Swap $R_i \leftrightarrow R_j$

2. Scale $kR_i$ ($k \neq 0$)

3. Add $R_i + nR_j \to R_i$

Example:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Add $2 \times$ Row 1 to Row 2:

$$B = \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix}$$

Subtracting $2 \times$ Row 1 from Row 2 returns $A$.

Conclusion: Statement C is True.

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Statement D: General Row Operations

Like column operations, if a row operation involves multiplying by zero or replacing a row with a constant that ignores the previous values, it is not reversible.

Counter-example:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Replace Row 1 with the zero row:

$$B = \begin{bmatrix} 0 & 0 \\ 3 & 4 \end{bmatrix}$$

Since $B$ is singular (determinant is 0) while $A$ was non-singular, we cannot perform an inverse operation to find $A$.

Conclusion: Statement D is False.

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Final Summary

• True Statements: B and C (Elementary operations are always reversible).

• False Statements: A and D (General operations can result in information loss).

 

Comments

Learned new things, thanks.

 

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clear explanation bro

 

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well explained

Answer 2

• A. False. General column operations need not be reversible.
  A column operation can reduce information, so getting back the original matrix may not be possible.
   
• B. True. Every elementary column operation is invertible, and its inverse is again an elementary column operation.
  So if $B$ is obtained from $A$ by elementary column operations, then $A$ can also be obtained from $B$ by elementary column operations.
   
• C. True. Exactly the same argument as B, but for rows.
   Every elementary row operation has an elementary inverse.
   
• D. False. General row operations need not be reversible, so the original matrix may not be recoverable.
   

Hence, the correct answer is B and C.

Answer 3

To Answer This Question Just know this :
1. Row operations : in these operations we need not to retain information of current row Like R2= R3+R1 so we cannot come back to previous R2 
2. Elementary Row Operations : in these operations we need to retain information.
R2'=R2-KR1

from transformed to get previous R2 we can do this 
R2=R2'+KR1
we can get information back.

Answer 4

Correct options: B, C

Answer 5

Answer : B is true

Answer 6

B and C are true