Q.Apply the given elementary transformation on each of the following matrices. $A = \begin{bmatrix} 5 & 4 \\ 1 & 3 \end{bmatrix}$, $C_1 \leftrightarrow C_2$; $B = \begin{bmatrix} 3 & 1 \\ 4 & 5 \end{bmatrix}$, $R_1 \leftrightarrow R_2$. What do you observe?

An elementary transformation is one of six basic, reversible operations performed on the rows or columns of a matrix. There are three row transformations, each with a column counterpart: (a) interchange two rows/columns, written $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_j$; (b) scalar multiplication of every entry of a row/column by a nonzero constant $k$, written $R_i \to kR_i$ or $C_i \to kC_i$; (c) scalar addition, where a multiple of one row/column is added entrywise to another, written $R_j \to R_j + kR_i$ or $C_j \to C_j + kC_i$. After transformation (a) or (c) the new matrix is generally not equal to the original, but the two are called equivalent matrices, written $A \sim B$, because each can be recovered from the other by a reverse elementary transformation. A key bookkeeping rule for transformation (c): in $R_j \to R_j + kR_i$, the row $R_i$ that supplies the multiple is left unchanged in the new matrix — only $R_j$ is overwritten (similarly for columns). Elementary transformations never change a matrix's order, and applying a row transformation never disturbs the effect of a previously-applied column transformation (and vice versa), because the two act on independent directions of the grid — this is why a matrix can be simplified using row and column operations in either order and reach the same equivalent matrix, provided the same set of operations is used. Elementary transformations are the working tool behind two major techniques of this chapter: reducing a matrix to row-echelon/triangular form, and computing the inverse of a nonsingular matrix without needing the determinant-based adjoint formula.