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 3×3 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 3×3 Matrix
Typically, there are 2 methods of assessing the determinant of a 3×3 matrix to employ as following
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General Method
In order to obtain the determinant of a 3×3 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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Shortcut Method
This is a clever trick to obtain the determinant of a 3×3 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
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There is a condition to attain a matrix determinant that is that the matrix should be a square matrix in order to compute it.
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There is no existence of the determinant of a non square matrix.
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Only determinants of square matrices are described by way of maths
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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|