Determinant of a 3 X 3 Matrix

Definition of Determinant of a 3 x 3 Matrix

In mathematics, a determinant is a value that is described for a square matrix. It is of crucial importance when solving systems of linear equations using a matrix. Determinants are the special numbers in matrices. Determinants are calculated from the square matrix. Likewise, the determinant of a 3 x 3 matrix is computed for a matrix with 3 rows and 3 columns, implying that the matrix must have an equal number of rows and columns. Those beings, so, let’s understand what the determinant of a matrix is. Imparting the knowledge that a matrix is an arrangement accommodating information of a linear transformation, and that this arrangement can comply with the coefficients of each variable in an equation system, we can literally describe the function of a determinant.

Mathematically, the determinant of 3x3 matrices is defined as

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Importance of Determinant in Linear Transformation

A determinant is all capable of scaling the linear transformation from the matrix. It can help you to attain the inverse of the matrix (given that there is one) and will further allow solving the systems of linear equations by initiating circumstances in which we can anticipate certain outcomes or features from the system (dependent on the determinant and the type of linear system). It also allows us to have familiarity with the fact if we may anticipate a unique solution, number of solutions (one or more than one) or none at all for the system.


Finding Determinant of a 3x3 Matrix

Typically, there are 2 methods of assessing the determinant of a 3x3 matrix to employ as following

  1. General Method

In order to obtain the determinant of a 3x3 matrix using the general method, break down the matrix into secondary matrices of shorter dimensions in a procedure referred to "expansion of the first row".

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  1. Shortcut Method

This is a clever trick to obtain the determinant of a 3x3 matrix that equips the calculation of a determinant of a large matrix by straightaway multiplying and subtracting or (adding) all of the information elements in their relevant module, without having to go all across the matrix expansion of the first row as well assessing the determinants of secondary matrices'.

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Uses of Determinant of a Matrix

The determinant of a matrix is most commonly used in calculus, advanced geometry and linear algebraic systems. Finding the determinant of a matrix becomes easier with few practice sessions. So, let’s get going with some practice problems.


Solved Examples

How to Find the Determinant of a 3×3 Matrices

Problem 1: Find the determinant of the 3×3 matrix below

Let’s say M = [2 -3 1]

          [2 0 -1]

          [1 4 5]

Solution 1:

In order to find out the determinant of the 3×3 matrix

We create here a set-up to enable you establish the correspondence between the generic elements of the formula and the elements of the real problem

 [M N O]                                         [2,-3, 1]

 [P Q R]   = 3 by 3 matrix with elements     [2, 0,-1]

[S T U]                                   [1, 4, 5]

Now, using the formula to find the determinant of a square matrix

   [M N O]                              

  det. [P Q R] = m. det [Q R; T U] - n. det [P R; S U] + O. det [P Q; S T]

      [S T U]                               

det [2,-3, 1; 2, 0,-1;1, 4, 5] = 2. det [ 0 -1; 4 5] - (-3) .det [ 2 -1; 15] + 1.det [2 0; 1 4]

Thus, we obtain

= 2 [0- [ -4} ] + 3 [ 10 - {-1}] + 1 [8-0]

= 2 {0+4} + 3 {10+1} +1 {8}

= 2 {4} + 3 {11} + {8}

Hence,

= 8 + 33 + 8

= 49


Problem 2:

Find out the matrix P as described below:

P= [2 -5 3; 0 7 -2; -1 4 1]

Solution 2:

Using the shortcut method gives us

det | P| = [ { 2 × 7 × 1) + { -5× -2 × -1} + { 3×0×4} - [{3 × 7 × -1} + { 2 × -2 × 4} +

{-5 ×0 × 1}]

Thus, we obtain

det | P| = ( 14 - 10 + 0) - ( 21 - 16 + 0) = 4 - (-37) = 41


Fun Facts

  • There is a condition to attain a matrix determinant that is that the matrix should be a square matrix in order to compute it.

  • There is no existence of the determinant of a non square matrix.

  • Only determinants of square matrices are described by way of maths

  • We generally write down matrices and their determinants in a similar way. For example the determinant of a matrix can be simply described as det P, det (P) or |P|

FAQ (Frequently Asked Questions)

1. What is A Matrix?

For the purpose of defining what a determinant of a matrix is, we need to learn about matrix. A matrix is a list of numbers arranged in an order inside a rectangular bracket. This list is generally termed as a rectangular array, and it renders a well-organized module to exhibit a "list" of elements of information.

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2. What is The Usefulness of A Matrix in Function of The Determinant?

A matrix narrates a linear transformation or linear chart, which is a form of transcription between 2 types of algebraic compositions, like vector fields. In such a way, we can obtain solutions of linear equations systems by characterizing a linear system as a matrix. The matrix characterization of a linear system is composed taking into account all of the variable coefficients found in the system. These coefficients are further used as element entries to build the rectangular array of a suitable size augmented matrix. In a matrix like this, the outcomes of each equation from the system will be positioned along the vertical line on the right hand side of which display the equal sign.

3. How to Find The Determinant of A Large Matrix?

The process to determine the determinant of a matrix having larger dimensions than 3x3 comply with the similar logic that you have seen this far. Employing the general method on a 5x5 matrix P, where its first (top) row adheres to the information elements P, Q, R, S, and T. However, you will have extended information on large expansions in other chapters.