What are pivots of a matrix?

by | Oct 3, 2023 | Featured, Linear Algebra

Pivots Of A Matrix

In our previous article, we discussed what the row echelon form of a matrix is. We learned that a matrix is said to be in the row echelon form if the following conditions hold:

1. The first non-zero element of a row is to the right of the first non-zero elements of the rows above.
2. Rows that contain all zeros are at the bottom of all rows that contain at least one non-zero element.

For example, the matrices given below are in the row echelon form.

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

When a matrix is in the row echelon form, the left-most non-zero element of each row is called a pivot. For example, for the given matrix A, 1 and 5 are the pivots.

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

For the matrix B, 1, 4, and 1 are pivots of the matrix.

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

And for the matrix C, 1 and 5 are the pivots.

C=\begin{bmatrix} 1 & 2 \\ 0 & 5 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}

 

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Amrita Mitra

Author

Ms. Amrita Mitra is an author, who has authored the books “Cryptography And Public Key Infrastructure“, “Web Application Vulnerabilities And Prevention“, “A Guide To Cyber Security” and “Phishing: Detection, Analysis And Prevention“. She is also the founder of Asigosec Technologies, the company that owns The Security Buddy.

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