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Elementary Operations on Matrices

Last Updated : 16 Jun, 2022

Matrices being the rectangular grid of numbers that consists of numbers carrying data as well as expressions of mathematical equations. Matrices are commonly used operation for computer engineering applications for obtaining approximate calculations. Besides, computing operations they are generally used for plotting graphs, Scientific studies, representation of data, and statistics in real life

Matrix is a branch of linear algebra that involves the systematic arrangement of numbers in rows and columns as per the linear equation in the form of a rectangular grid.

The great mathematician Arthur Cayley is the father of matrices who proposed the theory in 1858. It has dimensions like rows and columns for the arrangement of numbers. And each number involved in the matrix is known as the element of the matrix.

Types of Matrices

Null matrix: The matrix in which all the elements are zero is known as a null matrix or zero matrices. Generally, it is denoted by ‘0’. Then, if a_ij= 0 for all the elements of i and j

$0=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

Triangular matrix: The square matrix in which elements above or below the principal diagonal are triangular matrix. If the elements above the principal diagonal are zero then it is a lower triangular matrix and if the elements below the principal diagonal are zero it is an upper triangular matrix.

Take a look at below given lower and upper triangular matrix,

$A=\begin{bmatrix}1&0&0\\2&4&0\\3&5&6\end{bmatrix}$

$A=\begin{bmatrix}1&5&6\\0&2&4\\0&0&3\end{bmatrix}$

Column matrix: The matrix which only has one column is known as a column matrix. The order of the column matrix is always seen to be m×1.

$A=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}$

Row matrix: The matrix which only has one row is known as row matrix. The order of the matrix is always seen to be 1Xn.

$A=\begin{bmatrix}1&2&3&4\end{bmatrix}$

Horizontal matrix: The matrix with both rows and columns in order of m×n is a horizontal matrix. In the horizontal matrix number of columns needs to be greater than the number of rows (n>m).

$A=\begin{bmatrix}5&1&6&4\\2&1&3&2\end{bmatrix}$

Vertical matrix: The matrix with both rows and columns in order m×n is a vertical matrix. In the vertical matrix Number of rows needs to be greater than the number of columns (m>n).

$A=\begin{bmatrix}1&2\\2&5\\3&1\\4&1\end{bmatrix}$

Identity matrix: When all the elements of the principal diagonal are 1 in a matrix then it is said to be an identity matrix or unit matrix.

$I=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

Diagonal matrix: If all the elements in a square matrix are zero except the principal diagonal is known as a diagonal matrix.

$A=\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}$

Symmetric matrix: A square matrix which is a_ij=a_ji for all values of i and j is known as a symmetric matrix.

$A=\begin{bmatrix}1&2&3\\2&4&5\\3&5&2\end{bmatrix}$

Elementary Matrix Operations

Generally, there are three known elementary matrix operations performed on rows and columns of matrices. The operations performed on the rows are known as elementary matrix row operations. Whereas, the operations performed on columns are known as elementary matrix column operations.

The three different elementary matrix operations for rows are:

Interchange two Rows
Multiply a Row by a number
Adding one Row to another Row

And, the three elementary matrix operations for columns are:

Interchange Columns
Multiply a Column by a number
Adding one Column to another

Now, let’s look into how are these operations performed.

Elementary Matrix Row Operations

To perform the elementary row operations let suppose a matrix A_r×c that will be A_3×3.

Let $A=\begin{bmatrix}2&4&5\\4&8&3\\7&1&2\end{bmatrix}$

Interchange two Rows

This operation can be carried out by interchanging the position of any two rows of the matrix. It is indicated by R₁<=>R_2.

Interchanging the rows of matrix $A=\begin{bmatrix}2&4&5\\4&8&3\\7&1&2\end{bmatrix}$

Hence, R₁<=>R₂will be

$\begin{bmatrix}2&4&5\\4&8&3\\7&1&2\end{bmatrix}\leftrightarrow\begin{bmatrix}4&8&3\\2&4&5\\7&1&2\end{bmatrix}$

Here, the row 1 is replaced by row 2 and row 2 is replaced by 1. Whereas, the row 3 remains unchanged.

Multiplying a Row by a Number

This operation can be carried out by multiplying a row with a non-zero constant that will replace the elements of the row.

Lets multiply row 2 of the given matrix A= $\begin{bmatrix}2&4&5\\4&8&3\\7&1&2\end{bmatrix}$ by 2.

Hence, R₂<=>2R₂will be

$\begin{bmatrix}2&4&5\\4×2&8×2&3×2\\7&1&2\end{bmatrix}\leftrightarrow\begin{bmatrix}2&4&5\\8&16&6\\7&1&2\end{bmatrix}$

Here, the 2nd row is replaced by 2 times of itself.

Adding one Row to another

This operation can be performed by summing up anyone row with another one in the matrix. The remaining rows of the matrix remain unchanged. It can be indicated by R₁+R₂<=>R₂

Let’s sum up rows 1 and 3 to replace the elements of row 3 in the given matrix.

$\begin{bmatrix}2&4&5\\4&8&3\\7+2&1+4&2+5\end{bmatrix}\rightarrow\begin{bmatrix}2&4&5\\4&8&3\\9&5&7\end{bmatrix}$

Here, row 3 is replaced by the sum of rows 1 and 3. Whereas, row 1 and 2 remains unchanged.

Elementary Matrix Column Operations

To perform the elementary matrix column operation let us suppose a matrix A_r×c that will be A_3×3.

Let $\begin{bmatrix}1&2&4\\0&2&4\\0&3&5\end{bmatrix}$

Interchange two Columns

This operation can be carried out by interchanging the position of any two columns of the matrix. It is indicated by C₁<=>C₂.

Interchanging the columns of the matrix

$A=\begin{bmatrix}1&2&4\\0&2&4\\0&3&5\end{bmatrix}$

Hence, C₁<=>C₂ will be

$\begin{bmatrix}1&2&4\\0&2&4\\0&3&5\end{bmatrix}\leftrightarrow\begin{bmatrix}2&1&4\\2&0&4\\3&0&5\end{bmatrix}$

Here, the column 1 is replaced by column 2 and column 2 is replaced by 1. Whereas, the column 3 remains unchanged.

Multiplying a Column by a Number

This operation can be carried out by multiplying a column with a non-zero constant that will replace the elements of the column.

Let’s multiply column 2 of the given matrix

$A=\begin{bmatrix}1&2&4\\0&2&4\\0&3&5\end{bmatrix}$

Hence, 2C₂=>C₂ will be

$\begin{bmatrix}1&2×2&4\\0&2×2&4\\0&3×2&4\end{bmatrix}\rightarrow\begin{bmatrix}1&4&4\\0&4&4\\0&6&5\end{bmatrix}$

Here, the column 2 is replaced by 2 times of itself.

Adding one Column to another

This operation can be performed by summing up anyone column with another one in the matrix. The remaining columns of the matrix remain unchanged. It can be indicated by C₁+ C₂= C₂

Let’s sum up columns 1 and 2 to replace the elements of column 2 in the given matrix.

$A=\begin{bmatrix}1&2&4\\0&2&4\\0&3&5\end{bmatrix}$

Hence, C₁+C₂=C₂ will be

$\begin{bmatrix}1&2+1&4\\0&2+0&4\\0&3+0&5\end{bmatrix}\rightarrow\begin{bmatrix}1&3&4\\0&2&4\\0&3&5\end{bmatrix}$