Let a = (aij) be an n-rowed square matrix and I12 be the matrix obtained by interchanging the first and second rows of the n-rowed Identity matrix. Then AI12 is such that its firsta)Row the same as its second rowb)Row is the same as the second row of Ac)Column is the same as the second column Ad)Row is all zeroCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Question Analysis:
We are given a square matrix 'A' and we need to find the matrix 'AI12', obtained by interchanging the first and second rows of the n-rowed Identity matrix. We need to determine the properties of the resulting matrix 'AI12'.

Solution:
To solve this problem, let's break it down into steps:

Step 1: Write the given information:
- 'A' is an n-rowed square matrix.
- 'I12' is the matrix obtained by interchanging the first and second rows of the n-rowed Identity matrix.

Step 2: Understand the properties of the Identity matrix:
- The Identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
- It is denoted by 'I'.
- For example, if n = 3, then the Identity matrix is:
I = |1 0 0|
|0 1 0|
|0 0 1|

Step 3: Understand the operation of interchanging rows:
- Interchanging rows means swapping the elements of one row with the elements of another row.
- In this case, we are interchanging the first and second rows of the Identity matrix 'I' to obtain 'I12'.

Step 4: Calculate 'AI12':
- 'AI12' is the matrix obtained by multiplying 'A' with 'I12'.
- To calculate 'AI12', we need to multiply each element of 'A' with the corresponding element of 'I12'.
- Since 'I12' is obtained by interchanging the first and second rows of 'I', the resulting matrix will have the same elements as 'A', but with the first and second rows interchanged.
- Therefore, the resulting matrix 'AI12' will have the same elements as 'A', but with the second row as the first row and the first row as the second row.

Step 5: Determine the properties of 'AI12':
- From the above explanation, we can conclude that the resulting matrix 'AI12' will have its:
- First row the same as the second row of 'A'.
- Second row the same as the first row of 'A'.
- Third row (if applicable) the same as the third row of 'A'.
- And so on, for all the rows of 'A'.

Conclusion:
The correct answer is option 'C', which states that the column of 'AI12' is the same as the second column of 'A'. This is the correct interpretation of the properties of 'AI12' based on the given information and matrix operations.

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