Math Doubts

Determinant of a 2 × 2 matrix

Formula

$\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}$ $\,=\,$ $ad-bc$

Introduction

Let $M$ be a matrix. It has four elements $a$, $b$, $c$ and $d$. The four elements are arranged in a matrix form as follows.

$M$ $\,=\,$ $\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$

The matrix $M$ has two rows and two columns. Hence, it is called a $2 \times 2$ matrix. It is also called a square matrix.

Representation

According to the determinant of a matrix, the determinant of matrix $M$ is written in mathematical form as $det(M)$ or $|M|$

$(1).\,\,\,$ $det(M)$ $\,=\,$ $\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}$

$(2).\,\,\,$ $|M|$ $\,=\,$ $\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}$

You can follow any one of them for expressing the determinant of any $2$ by $2$ matrix in mathematical form.

Steps

There are three mathematical steps for finding the determinant of any two by two matrix.

  1. Find the product of diagonal elements. In this case, $a$ and $d$ are diagonal elements.
  2. Evaluate the product of anti-diagonal elements. In this case, $b$ and $c$ are anti-diagonal elements.
  3. Find the subtraction of the product of anti-diagonal elements from the product of diagonal elements for evaluating the determinant of any matrix of order $2$.

Therefore, the determinant of a square matrix of order two can be expressed in algebraic form as follows.

$\,\,\,\therefore\,\,\,\,\,\,$ $|M|$ $\,=\,$ $ad-bc$

Example

Find the determinant of matrix $P$ $\,=\,$ $\begin{bmatrix} 2 & -6 \\ 3 & 7 \\ \end{bmatrix}$

$\implies$ $|P|$ $\,=\,$ $\begin{vmatrix} 2 & -6 \\ 3 & 7 \\ \end{vmatrix}$

$\implies$ $|P|$ $\,=\,$ $2 \times 7 \,-\, (-6) \times 3$

$\implies$ $|P|$ $\,=\,$ $14 \,-\, (-18)$

$\implies$ $|P|$ $\,=\,$ $14+18$

$\,\,\,\therefore\,\,\,\,\,\,$ $|P|$ $\,=\,$ $32$

Thus, we can determine the determinant of any matrix of order two in matrices.

Math Doubts

A best free mathematics education website that helps students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved