How many elementary operations are possible on Matrices?

The correct option is 6.

An elementary operation on a matrix is an operation that changes the matrix's form while preserving important structural properties, such as the row space or column space. These operations are fundamental in linear algebra for tasks like finding the inverse of a matrix or solving systems of linear equations.

There are three types of elementary operations that can be performed, and each type can be applied to either the rows or the columns of a matrix. This results in a total of:
$$3\times 2=6$$
possible elementary operations.

Let's look at the three types of operations in detail:

1. Interchange of any two rows or columns:
We can swap any two rows, represented as:
$$R_i\leftrightarrow R_j$$
Or we can swap any two columns, represented as:
$$C_i\leftrightarrow C_j$$
This gives 2 operations (1 row operation + 1 column operation).

2. Multiplication of the elements of any row or column by a non-zero number:
We can multiply a row by a non-zero scalar $k$, represented as:
$$R_i\to k{R}_i$$
Or we can multiply a column by a non-zero scalar $k$, represented as:
$$C_i\to k{C}_i$$
This gives 2 operations (1 row operation + 1 column operation).

3. Addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number:
We can add a multiple of one row to another row, represented as:
$$R_i\to R_i+k{R}_j$$
Or we can add a multiple of one column to another column, represented as:
$$C_i\to C_i+k{C}_j$$
This gives 2 operations (1 row operation + 1 column operation).

Combining these, we have 3 row operations and 3 column operations, making a total of 6 elementary operations possible on matrices.