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Let $A=(a_{ij})$ be an $n$-rowed square matrix and $I_{12}$ be the matrix obtained by interchanging the first and second rows of the $n$-rowed Identity matrix. Then $AI_{12}$ is such that its first

  1. Row is the same as its second row
  2. Row is the same as the second row of $A$
  3. Column is the same as the second column of $A$
  4. Row is all zero
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Take a $2\times2$ matrix and multiply it with $\bigl(\begin{smallmatrix} 0 &1 \\ 1& 0 \end{smallmatrix}\bigr)$. You will get to see that the columns of the original matrix has interchanged. Hence option $(C)$ is correct.
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A permutation matrix, denoted as P, is a square matrix that represents a permutation of the rows or columns of an identity matrix.

When you multiply a permutation matrix on the left side of a matrix A (PA), it actually permutes the rows of matrix A according to the permutation represented by the permutation matrix P.

PA = A (with row of A swapped acc to permutation of P)

Similarly, when you multiply a permutation matrix on the right side of a matrix A (AP), it permutes the columns of matrix A according to the same permutation represented by the permutation matrix P.

AP = A (with columns of A swapped acc to permutation of P)

Here, $AI_{12}$   hence multiplying permutation matrix on right side of A and 1st row and 2nd rows of P are exchanged (exchanging 1st 2nd rows or exchanging 1st 2nd column of $I_{12}$ is equivalent).

Hence $AI_{12}$ will record exchanging 1st and 2nd columns of matrix A.

Answer:

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