Elementary Column Operations: How They Simplify And Transform Matrices Without Affecting Determinant And Eigenvalues

elementary column operations do not necessarily preserve rank

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Elementary column operations are operations that we perform on the columns of a matrix to simplify or transform the matrix without changing its properties such as determinant, eigenvalues, and rank. The three types of elementary column operations are (1) interchanging any two columns of a matrix, (2) multiplying any column of a matrix by a non-zero scalar and (3) adding a multiple of any column of a matrix to another column.

Although elementary column operations do not change the determinant and eigenvalues of a matrix, they do not necessarily preserve the rank of a matrix. The rank of a matrix is the number of linearly independent rows or columns of a matrix. When we perform elementary column operations, we change the linear combinations of the columns of a matrix. Thus, the operation may result in a column that is a linearly dependent combination of other columns, which would reduce the rank of the matrix.

For example, consider the matrix \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix}. Swapping the first and second columns, we get \begin{bmatrix} 2 & 1 & 3 \\ 3 & 2 & 4 \\ 4 & 3 & 5 \end{bmatrix}. Although the determinant is unchanged and the matrix is still invertible, the rank has decreased from 3 to 2. Similarly, multiplying the second column by 2 and subtracting it from the third column, we get \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & -2 \\ 3 & 4 & -1 \end{bmatrix}. Again, the determinant is unchanged, but the rank has decreased from 3 to 2.

In summary, elementary column operations are useful tools for simplifying and transforming matrices, but they do not necessarily preserve the rank of a matrix.

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