Elementary Row Operations

The Operations

In the context of Gaussian elimination, we resort to one of these three row operations:

  • Interchange any two rows of a matrix.
  • Add a multiple of a row to another row.
  • Multiply a row by a non-zero constant.

Property-1: Reversibility

All these operations are “reversible”. This means that if we start with a matrix \(A\) and perform a row operation to get the matrix \(B\), we can get back \(A\) by performing the opposite operation on \(B\). For example,

  • If we swap the first and second rows of \(A\) to get \(B\), we can swap the first and second rows of \(B\) to get \(A\).
  • If we add two times the first row to the second row of \(A\) to get \(B\), we can subtract two times the first row from the second row of \(B\) to get \(A\).
  • If we multiply the third row of \(A\) by three to get \(B\), we can divide the third row of \(B\) by three to get \(A\).

Property-2: Elementary matrices

Each elementary row operation is associated with a matrix. Let us consider the operation of interchanging two rows of the matrix \(A\) to get \(B\), where: \[ A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}, B = \begin{bmatrix} 4 & 5 & 6\\ 1 & 2 & 3\\ 7 & 8 & 9 \end{bmatrix} \] The operation is: \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \xrightarrow[]{R_1 \leftrightarrow R_2} \begin{bmatrix} 4 & 5 & 6\\ 1 & 2 & 3\\ 7 & 8 & 9 \end{bmatrix} \] This can be achieved by pre-multiplying \(A\) with the matrix \(E = \begin{bmatrix}0 & 1 & 0\\1 & 0 & 0\\0 & 0 & 1\end{bmatrix}\). Pre-multiply means to have \(E\) before \(A\). Recall that matrix multiplication is not commutative, that is, the order matters. You can verify that the following matrix multiplication comes through: \[ EA = B \] \(E\) is called an elementary matrix. Since the row operation is reversible, we can intuitively see that the matrix \(E\) is reversible or invertible. Thus, we have: \[ E^{-1} B = A \] Each row operation is associated with one such elementary matrix that is invertible. A sequence of row operations can therefore be seen as a string of matrix multiplications. If we perform three row operations on \(A\) with the corresponding elementary matrices being \(E_1, E_2, E_3\), then the resulting matrix is: \[ E_3 E_2 E_1 A \] Note the apparent change in order: \(3 \rightarrow 2 \rightarrow 1\). This is because, we first have \(E_1A\), then we move to \(E_2E_1A\) and finally move to \(E_3 E_2 E_1 A\).