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Elementary Matrix Calculator


Table of Content


The elementary matrix calculator determines the operations of rows or columns on the identity matrix.

What Is An Elementary Matrix?

“An elementary matrix is a square matrix that is obtained by performing the operations of elementary for columns and rows on an identity matrix”.

It differs from the identity matrix by one single elementary row operation. 

How To Operate The Calculator?

The calculator has a simple and user-friendly interface. So put the values in the tool and discover the operations on the identity matrices.  


  • Select the option of row or column elementary matrix
  • Put the size of the matrix in the given field 
  • Set the Pth row or column  
  • Enter the qth row or column 
  • Insert the values of a and b


  • The elementary matrices 
  • Complete calculations with steps 

Elementary Row or Column Operations:

The elementary row operation calculator performs the calculations by using matric multiplication using the square matrices. There are three types of elementary matrices that are considered for the row operations and respectively for the column operations. 

Row or Column Interchanging:

A row or column in the matrix can be interchanged with another. 

$$ R_i\rightarrow R_j $$

Row or Column Multiplication:

It is also referred to as a scaling matrix in which each element of a row in either column can be multiplied by the non-zero constants.

$$ K R_i\rightarrow R_i $$

Row or Column Addition:

It is the process of addition of the multiples of the row or columns to the others. 

$$ R_i + R_jR_i $$

Elementary Matrix Formula:

If you need to apply the simple row or column operations on the identity metrics then It is extremely helpful to use the elementary matrix calculator. This calculator simplifies matrix operations effortlessly including addition, subtraction, multiplication, or inverse.

So look at the below formulas:

For Rows:

$$ aR_p + bR_q\rightarrow R_q $$

For Columns:

$$ aC_p + bC_q\rightarrow C_q $$


Evaluate the elementary row matrices for the given values and verify these with the help of the matrix elementary row operations calculator. 

Given Data:

  • Factor a = 5
  • Factor b = 4
  • Matrix Size (n)= 4
  • Resultant Row (Rq) = 4
  • pth Row (Rp) = 3

$$ aR_p + bR_q -> R_q $$

As we know that n =4 means an identity matrix of 4x4 order as:

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

aRp = 5 x R3 (Rp is 3 rd Row)

bRq = 4 x R4 (Rq is 4 th Row)

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

$$ 5R_3 + 4R_4 = aR_p + bR_q $$

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

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

You can verify the results in seconds with the help of an elementary row operations matrix calculator. 


What Is The Difference Between Identity and Elementary Matrices?

Both are squared matrices in which identity matrices are the involuntary matrix equals to their own inverse. All the elements of the diagonal are ones and others are zeros. 

Elementary matrices actually derived from the identity matrices by performing row or column operations. 

Is Zero An Elementary Matrix?

No, zero is not an elementary matrix because it needs more than one-row matrix to get it from the identity matrix.

Is The Elementary Matrix An Inverse Matrix?

An elementary matrix is used to correspond to the row operations and every row operation is reversible. So we say that an elementary matrix also has an inverse matrix. 


From the source Elementary matrix, How elementary matrices act on other matrices. 

From the source Wikipedia: Elementary matrix, Elementary row operations

Sarah Taylor

I am a professional Chemist/Blogger & Content Writer. I love to research chemistry topics and help everyone learning Organic & Inorganic Chemistry and Biochemistry. I would do anything to spend vacations on a Hill Station.

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