Matrix Determinant – Laplace and Sarrus Method

N.B
some of matrix picture might be good since it was generated from https://www.symbolab.com/solver/matrix-vector-calculator , meanwhile some of them are my handwriting.

https://ringlayer.github.io

It has been so long since my last time  playing with matrix. The last time I used matrix is when I was study about input and output optimization using leontief matrix. Since it has been so long, I decided to write an interesting topic about matrix called determinant.

What is matrix determinant ? Here is a short  description from wikipedia:

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det, det A, or |A|

First of all, it’s only possible to find determinant of a matrix when a matrix has the same number of columns and rows. In order to calculate determinant of a matrix, there are many methods. There are 2 methods which I used frequently. They are : laplace method and sarrus method.

The most simple way to calculate a matrix determinant is when the matrix consists of 2 rows and 2 columns only.

Example we have this matrix called U matrix :

a

|U|  = (2 * 6) – (1 * 3) = 9

Sarrus Method

Based on sarrus method : “3×3 matrix determinant” is the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements. Suppose we have 3×3 matrix called A :

1

To calculate |A| of matrix A using sarrus method :

2

We added 2 columns from matrix beside our original matrix, so we get the sum of the products of three diagonal north-west to south-east lines of matrix elements :

(1 *  3 * 2 ) + (2 * 1 * 2) + (1 * 3 * 1)

Then we need to get the sum of the products of three diagonal south-west to north-east lines of elements :

3

(2 * 3 * 1) + (1 * 1 * 1) + (2 * 3 * 2)

Based on above informations, we got this:

|A| = ( (1 * 3 * 2 ) + (2 * 1 * 2) + (1 * 3 * 1) ) – ( (2 * 3 * 1) + (1 * 1 * 1) + (2 * 3 * 2) )

|A| = (6 + 4 + 3) – ( 6 + 1 + 12) =  -6

Laplace Method

Calculating determinant of 3×3 matrix using laplace is simple.

Based on literature, to calculate 3×3 matrix using laplace :

laplace

So the matrix is  splitted into 3 small matrices which  2×2 matrix, where a,b and c are constants. Those 3 small 2×2 matrices are permutation from this columns and rows:

4

the 2×2 matrices :

5

6

7

Back again to A matrix:

1

mark

a = 1 , b = 2, c = 1

We got 2×2 permutation matrices :

mark1

mark2

mark3

lap1

determinant of first matrix = 5

determinant of second matrix = 4

determinant of third matrix = -3

so we got : (1 * 5 ) – (2 * 4) + (1 * -3) = 5 – 8 – 3 = -6

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