Possible Operations on Matrices

There are six elementary operations that can be performed on matrices:

1. Interchange two rows or columns.
2. Multiply a row or column by a non-zero scalar.
3. Add a multiple of one row or column to another row or column.
4. Transpose (switch rows and columns).
5. Invert (find the inverse of) a matrix.
6. Multiply two matrices.

Explanation

1. Interchange two rows or columns: This operation involves swapping the positions of two rows or columns in a matrix. For example, if we have a matrix A with two rows, we can interchange the first and second rows to get a new matrix A' where the first row of A is now the second row of A' and vice versa.

2. Multiply a row or column by a non-zero scalar: This operation involves multiplying all the elements in a row or column of a matrix by a non-zero scalar. For example, if we have a matrix A and we multiply the second row by 2, we get a new matrix A' where all the elements in the second row of A are multiplied by 2.

3. Add a multiple of one row or column to another row or column: This operation involves adding a multiple of one row or column of a matrix to another row or column. For example, if we have a matrix A and we add 3 times the first row to the second row, we get a new matrix A' where the second row of A' is equal to the original second row of A plus 3 times the first row of A.

4. Transpose: This operation involves switching the rows and columns of a matrix. For example, if we have a matrix A with two rows and three columns, we can transpose it to get a new matrix A' with three rows and two columns.

5. Invert: This operation involves finding the inverse of a matrix. The inverse of a matrix A is denoted by A^-1 and is a matrix such that the product of A and A^-1 is the identity matrix.

6. Multiply two matrices: This operation involves multiplying two matrices together. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows and columns of the two matrices and then summing the products.