3×3 Matrix

A matrix that consists of the entries (or elements) in three rows and three columns is called a $3 \times 3$ matrix.

Introduction

A $3 \times 3$ matrix is a special matrix. It is very useful in mathematics. So, it is very important to study what a $3 \times 3$ matrix is. So, let’s learn the $3 \times 3$ matrix in detail.

The following matrix $M$ represents a $3 \times 3$ matrix.

$M$ $\,=\,$ ${\begin{bmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{bmatrix}}$

In a matrix of the order $3$, it has total nine elements and they are arranged in three rows and three columns. The arrangement of the nine entries forms a square shape inside the matrix. Hence, it is also called a square matrix of the order $3$.

Examples

The following three matrices are some examples for a $3 \times 3$ square matrix.

$(1).\,\,\,$ ${\begin{bmatrix} 4 & 7 & 2 \\ 5 & 3 & 8 \\ 1 & 4 & 6 \\ \end{bmatrix}}$

$(2).\,\,\,$ ${\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \\ \end{bmatrix}}$

$(3).\,\,\,$ ${\begin{bmatrix} 3 & 0 & 0 \\ 7 & 2 & 0 \\ 1 & 6 & 4 \\ \end{bmatrix}}$

Determinant

The determinant of a third order matrix is simply written as $\operatorname{det}(M)$ or $|M|$, and it is expressed in matrix form as follows.

$|M|$ $\,=\,$ ${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$

There is a special procedure for determining the determinant of a square matrix of the order $3$. So, let’s learn how to find the determinant of any $3 \times 3$ matrix in mathematics.