Math Doubts

Determinant of $3 \times 3$ matrix formula

Formula

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

According to the definition of the determinant of a matrix, a formula for the determinant of a 3 by 3 matrix can be derived in algebraic form by following four fundamental steps. The following mathematical expression represents the determinant of a square matrix of the order $3$ in algebraic form.

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

Now, let’s learn how to derive the determinant formula for the matrix of the order $3 \times 3$ by following the four steps.

Product form for the first element in first row

$\left|\begin{array}{c | c c} \color{red} e_{11} & e_{12} & e_{13} \\ \hline e_{21} & \color{blue} e_{22} & \color{blue} e_{23} \\ e_{31} & \color{blue} e_{32} & \color{blue} e_{33} \\ \end{array}\right|$

  1. The entry $e_{11}$ is selected from the first row.
  2. It is an element in the first row and first column. So, evaluate the $-1$ raised to the power of the sum of “the number of the row” and “the number of the column” for the element $e_{11}$. It is equal to $(-1)^{1+1}$.
  3. Evaluate the determinant of the elements by leaving the entries in the row and the column of the element $e_{11}$.
  4. Calculate the product by multiplying the entry $e_{11}$ with $(-1)^{1+1}$ and the determinant of the square matrix of order $2$.

$\implies$ $\left|\begin{array}{c | c c} \color{red} e_{11} & e_{12} & e_{13} \\ \hline e_{21} & \color{blue} e_{22} & \color{blue} e_{23} \\ e_{31} & \color{blue} e_{32} & \color{blue} e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{11} \times (-1)^{1+1} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$

Product form for the second element in first row

$\left|\begin{array}{c|c|c} e_{11} & \color{red} e_{12} & e_{13} \\ \hline \color{blue} e_{21} & e_{22} & \color{blue} e_{23} \\ \color{blue} e_{31} & e_{32} & \color{blue} e_{33} \\ \end{array}\right|$

  1. The element $e_{12}$ is chosen from the first row of the matrix.
  2. It is an entry in the first row and second column. Hence, find the $-1$ raised to the power of the sum of “the number of the row” and “the number of the column” for the chosen entry $e_{12}$. It means $(-1)^{1+2}$.
  3. Calculate the determinant of a matrix of the order $2$ by leaving the elements in the row and column of the selected entry $e_{12}$.
  4. Find the product by multiplying the entry $e_{12}$ with $(-1)^{1+2}$ and the determinant of $2 \times 2$ square matrix.

$\implies$ $\left|\begin{array}{c|c|c} e_{11} & \color{red} e_{12} & e_{13} \\ \hline \color{blue} e_{21} & e_{22} & \color{blue} e_{23} \\ \color{blue} e_{31} & e_{32} & \color{blue} e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{12} \times (-1)^{1+2} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$

Product form for the third element in first row

$\left|\begin{array}{cc|c} e_{11} & e_{12} & \color{red} e_{13} \\ \hline \color{blue} e_{21} & \color{blue} e_{22} & e_{23} \\ \color{blue} e_{31} & \color{blue} e_{32} & e_{33} \\ \end{array}\right|$

  1. Select the remaining element in the first row and it is $e_{13}$.
  2. This entry is an element in the first row and third column of the matrix. Therefore, calculate the $-1$ raised to the power of the sum of “the number of the row” and “the number of the column” for the chosen entry $e_{13}$, and it is written as $(-1)^{1+3}$.
  3. Find the determinant of a matrix of the order $2 \times 2$ by leaving the elements in the first row and third column.
  4. Evaluate the product by multiplying the entry $e_{13}$ with $(-1)^{1+3}$ and the determinant of the square matrix of order $2$.

$\implies$ $\left|\begin{array}{cc|c} e_{11} & e_{12} & \color{red} e_{13} \\ \hline \color{blue} e_{21} & \color{blue} e_{22} & e_{23} \\ \color{blue} e_{31} & \color{blue} e_{32} & e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{13} \times (-1)^{1+3} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{23} \\ \end{vmatrix}}$

Addition of the Products of first row entries

Finally, add the products for calculating the determinant of a square matrix of the order $3$.

${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11} \times (-1)^{1+1} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1)^{1+2} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times (-1)^{1+3} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$

Now, let’s evaluate the product in each term in the right hand side expression.

$=\,\,\,$ $e_{11} \times (-1)^{2} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1)^{3} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times (-1)^{4} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$

$=\,\,\,$ $e_{11} \times 1 \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1) \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times 1 \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$

$\therefore\,\,\,$ ${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,-\,$ $e_{12} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$

It can used as a formula for calculating the determinant of a third order square matrix. It can also be simplified further by the determinant of a second order matrix.

$=\,\,\,$ $e_{11} \times \Big(e_{22} \times e_{33}-e_{23} \times e_{32}\Big)$ $\,-\,$ $e_{12} \times \Big(e_{21} \times e_{33}-e_{23} \times e_{31}\Big)$ $\,+\,$ $e_{13} \times \Big(e_{21} \times e_{32}-e_{22} \times e_{31}\Big)$

$=\,\,\,$ $e_{11} \times \Big(e_{22}e_{33}-e_{23}e_{32}\Big)$ $\,-\,$ $e_{12} \times \Big(e_{21}e_{33}-e_{23}e_{31}\Big)$ $\,+\,$ $e_{13} \times \Big(e_{21}e_{32}-e_{22}e_{31}\Big)$

Each multiplying factor can be distributed to the difference of the terms in the expression by the distributive property of multiplication over subtraction.

$=\,\,\,$ $e_{11} \times e_{22}e_{33}$ $\,-\,$ $e_{11} \times e_{23}e_{32}$ $\,-\,$ $e_{12} \times e_{21}e_{33}$ $\,+\,$ $e_{12} \times e_{23}e_{31}$ $\,+\,$ $e_{13} \times e_{21}e_{32}$ $\,-\,$ $e_{13} \times e_{22}e_{31}$

$=\,\,\,$ $e_{11}e_{22}e_{33}$ $\,-\,$ $e_{11}e_{23}e_{32}$ $\,-\,$ $e_{12}e_{21}e_{33}$ $\,+\,$ $e_{12}e_{23}e_{31}$ $\,+\,$ $e_{13}e_{21}e_{32}$ $\,-\,$ $e_{13}e_{22}e_{31}$

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

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved