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Elementary Matrix Calculator

Enter the values in the required fields properly, and the tool will calculate the elementary matrix accordingly.

I want to find

Size of Matrix (n)

Pth Row (R_p)

Factor "a" to be multiplied by R_p

Row q that Receives Result(R_q)

Factor "b" to be multiplied by R_q

Related

The Elementary Matrix Calculator allows you to compute an elementary matrix by applying a single row or column operation to an identity matrix quickly and accurately.

What Is an Elementary Matrix?

An elementary matrix is a square matrix created by performing a single row or column operation on an identity matrix. There are three fundamental types of operations used to generate an elementary matrix:

Swapping two rows or two columns of the identity matrix
Multiplying a row or column by a non-zero scalar
Adding a multiple of one row or column to another row or column

If manual calculations seem cumbersome, the elementary matrix calculator can simplify the process for you.

Elementary Matrix Operations:

Row Operation:

$$ aR_p + bR_q \to R_q $$

Column Operation:

$$ aC_p + bC_q \to C_q $$

Using the calculator, you can instantly apply any row or column operation on an identity matrix.

Example:

Determine the elementary matrix for the following parameters:

$a = 2$

$b = 5$

$R_p = 1$

$R_q = 3$

Given:

Matrix Size (n) = 3

Pth Row (Rp) = 1

qth Row (Rq) = 3

Factor a = 2

Factor b = 5

Solution:

The identity matrix of size 3 is:

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

The row operation formula is:

$$ aR_p + bR_q \to R_q $$

Apply the operation:

2R1 + 5R3 = New R3

Intermediate matrix:

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

Final elementary matrix:

$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{bmatrix}$$

The calculator provides the same result instantly, saving manual computation time.

How the Elementary Matrix Calculator Works:

Using the calculator is straightforward. Input the following details:

Input:

Select "Row" or "Column" operation
Enter the matrix size
Enter the Pth row or Pth column
Enter the qth row or qth column
Enter values for "a" and "b"
Click "Calculate"

Output:

The resulting elementary matrix
Step-by-step explanation of the calculation

FAQs:

What is the difference between an elementary matrix and an identity matrix?

An identity matrix consists only of 1s on the diagonal and 0s elsewhere. An elementary matrix is derived from the identity matrix by performing one row or column operation, so it may contain other non-zero entries.

Is an elementary matrix always square?

Yes, by definition, an elementary matrix is always a square matrix.

Do row and column operations yield the same elementary matrix?

No, performing a row operation creates a different elementary matrix than performing a column operation. The calculator can handle both types efficiently.

References:

Source: Statlect.com - Elementary Matrix, Definition: Wikipedia - Elementary Matrix

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