Elementary Operations on Matrices

Elementary Operations on Matrices are the operations performed on the rows and columns of the matrix that do not change the value of the matrix. A matrix is a way of representing numbers in the form of an array, i.e. the numbers are arranged in the form of rows and columns. In a matrix, the rows and columns contain all the values in the cell. We represent a matrix as [A]_m×n where A is a matrix m is the number of rows in the matrix and n is the number of columns of the matrix.

In this article, we will learn about Matrix, Types of Matrices, Elementary Operations on Matrices, Problems, and others in detail.

What are Matrices?

A matrix is a way of representing the numbers in the form of rows and columns and all the numbers are represented in the cells of this matrix. We represent the matrix as [A]_m×n and m represents the number of rows and n represents the number of columns. There are various types of matrices that we are going to learn further in this article.

Types of Matrices

Various types of Matrices are,

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

O~=~\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
• Multipling 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 Mumber
• 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}

Interchanging 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<=>R₂  will 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 <=>2R₂ will 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}

Interchanging 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}

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

Read More,,

• Matrix Operation
• Matrix Multiplication
• Adjoint of a Matrix

Problems on Elementary Operation of Matrices

Problem 1: Perform Operation R₁ <=> R₂ on the given matrix.

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

Solution:

R₁ <=> R₂  means to interchange Rows 1 and 2 with each other.

So, Matrix 

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

In given operation, row 1 replaces row 2 and row 2 replaces row 1. Whereas, row 3 remains unchanged.

Problem 2: Perform operation C₂ <=> C₃  on the given matrix.

A=\begin{bmatrix}2&3&0\\4&5&9\\6&6&1\end{bmatrix}  

Solution:

C₂ <=> C₃ means to interchange columns 2 and 3 with each other.

So Matrix becomes,

A~=~\begin{bmatrix}2&3&0\\4&5&9\\6&6&1\end{bmatrix}~\leftrightarrow~\begin{bmatrix}2&0&3\\4&9&5\\6&1&6\end{bmatrix}

In given operation, column 2 replaces 3 and column 3 replaces 2. Whereas, column 1 remains unchanged.

Problem 3: Perform the row operation 3R₁ => R₁ on given matrix.

\begin{bmatrix}2&4&6\\0&9&1\\3&5&6\end{bmatrix}

Solution:

3R₁ => R₁ means to multiply row 1 with a non-zero constants that is 3 to replace the elements of row 1.

A~=~\begin{bmatrix}2×3&4×3&6×3\\0&9&1\\3&5&6\end{bmatrix}~\rightarrow~\begin{bmatrix}6&12&18\\0&9&1\\3&5&6\end{bmatrix}

In given operation 3 times of 1 row replaces elements of row 1. Whereas, row 2 and 3 remains constant.

Problem 4: Perform the column operation 2C₂ => C₂ on the given matrix.

\begin{bmatrix}7&4&8\\2&6&5\end{bmatrix}

Solution:

2C₂ => C₂ means to multiply column 2 with a non-zero constant that is 2 to replace the elements of column 2.

A~=~\begin{bmatrix}7&4×2&8\\2&6×2&5\end{bmatrix}~\rightarrow~\begin{bmatrix}7&8&8\\2&12&5\end{bmatrix}

In given operation 2 times of column 2 replaces the elements of column 2. Whereas, column 1 remains unchanged.

Problem 5: Perform the row operation R₁ + R₂ => R₂  on the given matrix.

\begin{bmatrix}3&4&6\\2&5&3\end{bmatrix}

Solution:

R₁ + R₂ => R₂ means to replace the elements of row 2 with the sum of rows 1 and 2.

\begin{bmatrix}3&4&6\\2+3&5+4&3+6\end{bmatrix}\rightarrow\begin{bmatrix}3&4&6\\5&9&9\end{bmatrix}      

In given operation elements of row 2 is replaced by the sum of rows 1 and 2.

Problem 6: Perform the column operation C₁ + C₃ => C₃ on the given matrix.

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

Solution:

C₁ + C₃ => C₃ means to replace elements of column 3 with the sum of columns 1 and  3.

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

In the given operation elements of column 3 is replaced by the sum of columns 1 and 3.

Practice problems

1: Row Addition: Add 2 times the first row to the second row in the matrix:

\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}​

2: Row Multiplication: Multiply the second row by -3 in the matrix:

\begin{pmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{pmatrix}​

3: Row Switching: Switch the first and third rows in the matrix:

\begin{pmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 3 & 3 & 3 \end{pmatrix}​

4: Row Addition: Subtract the second row from the third row in the matrix:

\begin{pmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 3 & 6 & 9 \end{pmatrix}​

5: Row Multiplication: Multiply the third row by 1/2 in the matrix:

\begin{pmatrix} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 1 & 2 & 6 \end{pmatrix}​

6: Row Switching: Switch the second and third rows in the matrix:

\begin{pmatrix} 1 & 5 & 3 \\ 4 & 0 & 2 \\ 2 & 7 & 1 \end{pmatrix}​​

Summary

Elementary operations on matrices are fundamental manipulations used to solve linear systems, find matrix inverses, and perform other matrix-related calculations. These operations include row switching, row multiplication, and row addition.

1. Row Switching: Interchanging two rows of a matrix. This operation changes the order of the rows but not the solutions to the linear system represented by the matrix.
2. Row Multiplication: Multiplying all elements of a row by a non-zero scalar. This operation scales the row but preserves the relationships between the rows.
3. Row Addition: Adding a multiple of one row to another row. This operation is used to create zeros in specific positions to simplify the matrix and solve linear systems.