Trick to calculate determinant of a 3x3 matrix

Introduction to Determinants

Linear Algebra

It is one of the most important topics in the Engineering Mathematics Gate syllabus. Finding the determinant of a matrix is one of the most important problems in Linear Algebra. Finding the Determinant of a matrix is required for finding the inverse of a matrix, determining whether vectors are linearly independent or not, etc.

Determinant

A Determinant is a special number that can be calculated from a matrix by applying some predefined rules to it. In linear Algebra, the Determinant is a scalar value that is a function of entries of a square matrix. So, if we find the determinant of a Square Matrix A then it will give a scalar value that can be computed from its elements according to specific rules.

Determinant of a 3 X 3 Matrix

Let us consider an example of a 3X3 matrix and its determinant be A, then A can be calculated as given below.

where,

The determinant of a 3x3 matrix involves computing the sum of the products of its elements and the corresponding submatrix determinants, following the sign convention. This traditional method of finding the determinant of a square matrix is time-consuming and not efficient way of doing it . So we also have another method or we can say trick to find determinants faster than this traditional method.

Example :

A = 1( 5*9 – 6*8) – 2(4*9 – 6*7) + 3(4*8 – 5*7)
A = 1(45 – 48) – 2(36 – 42) + 3(32 – 35)
A = 1*(-3) – 2*(-6) + 3*(-3)
A = -3 + 12 – 9
A = 0 

The above traditional method consumes a lot of time especially when you are solving some complex problem. The below shown isa faster way of solving the determinant of a matrix.

Trick to Calculate the Determinant of the 3X3 matrix

Write the matrix as :

Calculate the determinant:

1. It can be done by calculating the sum of the products of diagonal elements from top of left to bottom of right

                                                              x = (a * e * i) + (b * f * g) + (c * d * h)

2. Then do the same from opposite side , the sum of the products of the diagonals from the bottom left to the top right.

                                                             y = (c * e * g) + (a * f * h) + (b * d * i)

3. Then do subtraction of x and y to get the determinant of matrix A.

det(A) = x-y

Let's take an example of Matrix A :

1. Find the sum of product of diagonals from left top to bottom right

                                                               x = (1 * 5 * 9) + (2 * 6 * 7) + (3 * 4 * 8)
                                                                 x = 45 + 84 +  96
                                                                 x = 225

2. Then find the opposite sum of product of diagonals from bottom left to top right

                                                                 y = (3 * 5 * 7) + (1 * 6 * 8) + (2 * 4 * 9)
                                                                   y = 105 + 48 +  72
                                                                   y = 225

3. Now subtract the x and y to find the determinant of the matrix A:

                                                                          det(A)= x - y
                                                                            det(A)=225−225
                                                                            det(A)=0

You can write the same above trick in this form as well :-

Here,

So, from the above matrix, we can write,

i = (3*4*8) = 96     x = (2*4*9) = 72
j = (1*5*9) = 45     y = (3*5*7) = 105
k = (2*6*7) = 84     z = (1*6*8) = 48

A = (i + j + k) – (x + y + z) = (96 + 45 + 84) – (72 + 105 + 48) 
  = (225 – 225) 
  = 0 

So, by following the above we can calculate the determinant of a matrix easily. It requires practice to change our method of calculation from traditional method to efficient one but it is worth practicing,

: - This method works only for (3, 3) matrix.