Elementary Row/Column Operations on a Matrix

Elementary Row/Column Operations on a Matrix refers to performing Elementary Algebraic Operations on Rows or Columns of Matrices with the objective of Reducing all the Vectors/Co-vectors in the Matrix to Mutually Orthogonal Identity Vectors/Co-Vectors or Reducing the Matrices to Row Echelon/Column Echelon Form.
Elementary Row/Column Operations on a Matrix can be used for following
1. Solving System of Linear Equations.
2. Finding Column Space, Row Space, NULL Space and Orthogonal Space of the Matrix.
3. Determining whether the Vectors that make up the Matrix are Linearly Dependent or Independent and Finding the Dependency.
4. Determining the Rank of Matrix.
5. Determining the Inverse of a Non-Singular Matrix.
6. Performing Matrix Factorization through LU /PLU /LUP Decomposition.
Following are the Elementary Row Operations that can be performed on Matrices
1. \(\mathbf{R_i \leftrightarrows R_j}\): Interchanging/Swapping all the elements \(a_i\) of \(i^{th}\) Row \(R_i\) with corresponding elements \(a_j\) of \(j^{th}\) Row \(R_j\) (belonging to the same Column as \(a_i\)).
2. \(\mathbf{R_i \leftarrow k \times R_i}\): Multiplying all the elements \(a_i\) of \(i^{th}\) Row \(R_i\) with a Non-Zero Constant \(k\).
3. \(\mathbf{R_i \leftarrow k \times R_i + l \times R_j}\): Replacing each element \(a_i\) of \(i^{th}\) Row \(R_i\) with \(k \times a_i + l\times a_j\), where \(k\) and \(l\) are Non-Zero Constants and \(a_j\) is element of \(j^{th}\) Row \(R_j\) (belonging to the same Column as \(a_i\)).
Similarly, following are the Elementary Column Operations that can be performed on Matrices.
1. \(\mathbf{C_i \leftrightarrows C_j}\): Interchanging/Swapping all the elements \(a_i\) of \(i^{th}\) Column \(C_i\) with corresponding elements \(a_j\) of \(j^{th}\) Column \(C_j\) (belonging to the same Row as \(a_i\)).
2. \(\mathbf{C_i \leftarrow k \times C_i}\): Multiplying all the elements \(a_i\) of \(i^{th}\) Column \(C_i\) with a Non-Zero Constant \(k\).
3. \(\mathbf{C_i \leftarrow k \times C_i + l \times C_j}\): Replacing each element \(a_i\) of \(i^{th}\) Column \(C_i\) with \(k \times a_i + l\times a_j\), where \(k\) and \(l\) are Non-Zero Constants and \(a_j\) is element of \(j^{th}\) Column \(C_j\) (belonging to the same Row as \(a_i\)).
Performing Row Operations on a Matrix is similar to Performing Column Operations on the Transpose of the Matrix and vice-versa.
To demonstrate the Algorithm for Performing Elementary Row/Column Operations, let's consider a \(M \times N\) Matrix \(A\) having elements \(a_{ij}\) as following

\(A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22}& a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(1)
The following are the Steps to Perform Elementary Row Operations in the Matrix \(A\) given in equation (1) above either to Convert the Matrix \(A\) to a Row Echelon Matrix or to Convert the Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors.
1. Interchange/Swap the 1st Row with the Rows below till the Element \(a_{11}\) becomes Non-Zero, i.e. Perform the following Row Operations
  
  \(R_1 \leftrightarrows R_k\)
  
  where \(k > 1\)
2. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors, Perform the Row Operations
  
  \(R_1 \leftarrow \frac{R_1}{a_{11}}\)
  
  i.e. Divide all elements of the 1st Row with Element \(a_{11}\) so that the Element \(a_{11}\) becomes 1 as given in the following
  
  \(A=\begin{bmatrix} 1 & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22}& a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(2)
  
  Please note that the Elements \(a_{12}, a_{13}, ... , a_{1n}\) will get updated accordingly.
  
  Please note that This Step Is Not Required and Must Not Be Performed if the goal is to Convert Matrix \(A\) to Row Echelon Matrix.
3. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors, Perform following Row Operation on every \(k^{th}\) Row (where \(k \neq 1\)) such that 1st Column of Every Row (except for the 1st Row) becomes 0
  
  \(R_k \leftarrow R_k - a_{k1} \times R_1\)
  
  \(A=\begin{bmatrix} 1 & a_{12} & a_{13} & ... & a_{1n}\\ 0 & a_{22}& a_{23} & ... & a_{2n} \\ 0 & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(3)
  
  If the goal is to Convert Matrix \(A\) to a Row Echelon Matrix, Perform following Row Operation on every \(k^{th}\) Row (where \(k > 1\)) such that 1st Column of Every Row (except for the 1st Row) becomes 0
  
  \(R_k \leftarrow R_k - \frac{a_{k1}}{a_{11}} \times R_1\)
  
  \(A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ 0 & a_{22}& a_{23} & ... & a_{2n} \\ 0 & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(4)
4. Interchange/Swap the 2nd Row with the Rows below till the Element \(a_{22}\) becomes Non-Zero, i.e. Perform the following Row Operations
  
  \(R_2 \leftrightarrows R_k\)
  
  where \(k > 2\)
5. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors, Perform the Row Operations
  
  \(R_2 \leftarrow \frac{R_2}{a_{22}}\)
  
  i.e. Divide all elements of the 2nd Row with Element \(a_{22}\) so that the Element \(a_{22}\) becomes 1 as given in the following
  
  \(A=\begin{bmatrix} 1 & a_{12} & a_{13} & ... & a_{1n}\\ 0 & 1 & a_{23} & ... & a_{2n} \\ 0 & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(5)
  
  Please note that the Elements \(a_{23}, a_{24}, ... , a_{2n}\) will get updated accordingly.
  
  Please note that This Step Is Not Required and Must Not Be Performed if the goal is to Convert Matrix \(A\) to Row Echelon Matrix.
6. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors, Perform following Row Operation on every \(k^{th}\) Row (where \(k \neq 2\)) such that 2nd Column of Every Row (except for the 2nd Row) becomes 0
  
  \(R_k \leftarrow R_k - a_{k2} \times R_2\)
  
  \(A=\begin{bmatrix} 1 & 0 & a_{13} & ... & a_{1n}\\ 0 & 1 & a_{23} & ... & a_{2n} \\ 0 & 0 & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(6)
  
  If the goal is to Convert Matrix \(A\) to a Row Echelon Matrix, Perform following Row Operation on every \(k^{th}\) Row (where \(k > 2\)) such that 2nd Column of Every Row greater than the 2nd Row becomes 0
  
  \(R_k \leftarrow R_k - \frac{a_{k2}}{a_{22}} \times R_2\)
  
  \(A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ 0 & a_{22} & a_{23} & ... & a_{2n} \\ 0 & 0 & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(7)
7. If the goal is to Convert Matrix \(A\) to a Row Echelon Matrix, Repeat the above steps for All Subsequent Rows till you have a Row Echelon Matrix.
  
  If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Vectors, Repeat the above steps for All Subsequent Rows till you have only 1's or 0's in the Main Diagonal of the Matrix. In such case After Reducing a Matrix using Elementary Row Operations, the Marix can have following 2 Kinds of Vectors
  1. Atleast One or More Identity Vectors (i.e. Vectors having 1 as One of the Components and 0 as Other Components).
  2. Zero or More Vectors that are Not Identity Vectors.
The following are the Steps to Perform Elementary Column Operations in the Matrix \(A\) given in equation (1) above either to Convert the Matrix \(A\) to a Column Echelon Matrix or to Convert the Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors.
1. Interchange/Swap the 1st Column with the Columns on its Right till the Element \(a_{11}\) becomes Non-Zero, i.e. Perform the following Column Operations
  
  \(C_1 \leftrightarrows C_k\)
  
  where \(k > 1\)
2. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors, Perform the Column Operations
  
  \(C_1 \leftarrow \frac{C_1}{a_{11}}\)
  
  i.e. Divide all elements of the 1st Column with Element \(a_{11}\) so that the Element \(a_{11}\) becomes 1 as given in the following
  
  \(A=\begin{bmatrix} 1 & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22}& a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(8)
  
  Please note that the Elements \(a_{21}, a_{31}, ... , a_{m1}\) will get updated accordingly.
  
  Please note that This Step Is Not Required and Must Not Be Performed if the goal is to Convert Matrix \(A\) to Column Echelon Matrix.
3. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors, Perform following Column Operation on every \(k^{th}\) Column (where \(k \neq 1\)) such that 1st Row of Every Column (except for the 1st Column) becomes 0
  
  \(C_k \leftarrow C_k - a_{1k} \times C_1\)
  
  \(A=\begin{bmatrix} 1 & 0 & 0 & ... & 0\\ a_{21} & a_{22}& a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(9)
  
  If the goal is to Convert Matrix \(A\) to a Column Echelon Matrix, Perform following Column Operation on every \(k^{th}\) Column (where \(k > 1\)) such that 1st Row of Every Column (except for the 1st Column) becomes 0
  
  \(C_k \leftarrow C_k - \frac{a_{1k}}{a_{11}} \times C_1\)
  
  \(A=\begin{bmatrix} a_{11} & 0 & 0 & ... & 0\\ a_{21} & a_{22}& a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(10)
4. Interchange/Swap the 2nd Column with the Columns on its Right till the Element \(a_{22}\) becomes Non-Zero, i.e. Perform the following Column Operations
  
  \(C_2 \leftrightarrows C_k\)
  
  where \(k > 2\)
5. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors, Perform the Column Operations
  
  \(C_2 \leftarrow \frac{C_2}{a_{22}}\)
  
  i.e. Divide all elements of the 2nd Column with Element \(a_{22}\) so that the Element \(a_{22}\) becomes 1 as given in the following
  
  \(A=\begin{bmatrix} 1 & 0 & 0 & ... & 0\\ a_{21} & 1 & a_{23} & ... & a_{2n} \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(11)
  
  Please note that the Elements \(a_{32}, a_{42}, ... , a_{m2}\) will get updated accordingly.
  
  Please note that This Step Is Not Required and Must Not Be Performed if the goal is to Convert Matrix \(A\) to Column Echelon Matrix.
6. If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors, Perform following Column Operation on every \(k^{th}\) Column (where \(k \neq 2\)) such that 2nd Row of Every Column (except for the 2nd Column) becomes 0
  
  \(C_k \leftarrow C_k - a_{2k} \times C_2\)
  
  \(A=\begin{bmatrix} 1 & 0 & 0 & ... & 0\\ 0 & 1 & 0 & ... & 0 \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(12)
  
  If the goal is to Convert Matrix \(A\) to a Column Echelon Matrix, Perform following Column Operation on every \(k^{th}\) Column (where \(k > 2\)) such that 2nd Row of Every Column greater than the 2nd Column becomes 0
  
  \(C_k \leftarrow C_k - \frac{a_{2k}}{a_{22}} \times C_2\)
  
  \(A=\begin{bmatrix} a_{11} & 0 & 0 & ... & 0\\ a_{21} & a_{22}& 0 & ... & 0 \\ a_{31} & a_{32}& a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\) ...(13)
7. If the goal is to Convert Matrix \(A\) to a Column Echelon Matrix, Repeat the above steps for All Subsequent Rows till you have a Column Echelon Matrix.
  
  If the goal is to Convert Matrix \(A\) to a Matrix containing Mutually Orthogonal Identity Co-Vectors, Repeat the above steps for All Subsequent Columns till you have only 1's or 0's in the Main Diagonal of the Matrix. In such case After Reducing a Matrix using Elementary Column Operations, the Marix can have following 2 Kinds of Co-Vectors
  1. Atleast One or More Identity Co-Vectors (i.e. Co-Vectors having 1 as One of the Components and 0 as Other Components).
  2. Zero or More Co-Vectors that are Not Identity Co-Vectors.

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