Cara Menghitung Determinan Matriks

>Hello Sohib EditorOnline, in this article we will discuss in detail about how to calculate the determinant of a matrix. Determinant is an important concept in linear algebra and finding it can be crucial in various applications of mathematics, physics and engineering. So, let’s get started with understanding the basics first.

What is Determinant?

Determinant is a value that can be calculated from a square matrix. It is represented by the symbol ‘det’. A square matrix has an order that represents the number of rows or columns in the matrix. For example, a matrix with 3 rows and 3 columns is called a 3×3 matrix.

The value of determinant of a matrix is used to determine various properties of the matrix such as its invertibility, rank, characteristic equation, etc. It is an important concept in linear algebra that finds applications in various fields of science and engineering.

Properties of Determinant

Before we move on to discuss how to calculate determinant, let’s first discuss some of its properties that will be useful in the calculation process.

Property	Formula
Property 1	If two rows of a matrix are interchanged, the determinant changes its sign.
Property 2	If any row of a matrix is multiplied by a constant ‘k’, the determinant is multiplied by ‘k’.
Property 3	If one row of a matrix is expressed as a linear combination of the other two rows, the determinant becomes zero.
Property 4	If two rows of a matrix are identical, the determinant is zero.

Calculating Determinant of a 2×2 Matrix

Let’s start with calculating the determinant of a 2×2 matrix. A 2×2 matrix is a matrix with 2 rows and 2 columns. It can be represented as:

a	b
c	d

The determinant of a 2×2 matrix can be calculated using the following formula:

ad – bc

Let’s take an example to understand it better. Suppose we have a 2×2 matrix:

2	3
4	5

The determinant of this matrix can be calculated as:

2	3
4	5

Determinant = (2 x 5) – (3 x 4) = 10 – 12 = -2

So, the determinant of this matrix is -2.

Calculating Determinant of a 3×3 Matrix

Calculating the determinant of a 3×3 matrix is a bit more complicated than a 2×2 matrix. However, it can be done using various methods such as cofactor expansion, row reduction, etc. Here, we will discuss the most commonly used method – cofactor expansion.

Cofactor Expansion Method

The determinant of a 3×3 matrix can be calculated using the following formula:

|A| = a11C11 + a12C12 + a13C13

Where:

a11, a12, a13	are the elements of the first row of the matrix.
C11, C12, C13	are the cofactors of the elements a11, a12, a13 respectively.

The cofactor of an element in a matrix can be calculated as:

Cij = (-1)i+j|Mij|

Where:

i, j	are the row and column numbers of the element for which we are calculating the cofactor.
|Mij|	is the determinant of the matrix obtained by deleting the ith row and jth column from the original matrix.

Let’s take an example to understand it better. Suppose we have a 3×3 matrix:

1	2	3
4	5	6
7	8	9

The determinant of this matrix can be calculated as:

|A| = 1C11 – 2C12 + 3C13

Let’s calculate the cofactors of the elements of the first row first:

C11 = (-1)1+1|M11| = (-1)2|5 6| = -5

C12 = (-1)1+2|M12| = (-1)3|4 6| = 6

C13 = (-1)1+3|M13| = (-1)4|4 5| = -4

Where |M₁₁| is the determinant of the matrix obtained by deleting the 1^st row and 1^st column from the original matrix.

|M₁₁| = |5 6| = (5 x 9) – (6 x 8) = 15 – 48 = -33

Similarly,

|M₁₂| = |4 6| = (4 x 9) – (6 x 7) = 36 – 42 = -6

|M₁₃| = |4 5| = (4 x 8) – (5 x 7) = 32 – 35 = -3

Now, substituting the values in the formula,

|A| = 1 x (-5) – 2 x 6 + 3 x (-4) = -5 – 12 – 12 = -29

So, the determinant of this matrix is -29.

Conclusion

In this article, we learned about the concept of determinant, its properties and different methods to calculate it. Determinant is an important concept in linear algebra and its calculation has various applications in science and engineering. So, it is important to have a good understanding of this concept. We hope this article was helpful in understanding how to calculate the determinant of a matrix.

FAQ

Q1. Can the determinant of a non-square matrix be calculated?

No, the determinant can only be calculated for a square matrix.

Q2. Is the determinant of a matrix unique?

Yes, the determinant of a matrix is unique.

Q3. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative, positive or zero.

Q4. Are there any other methods to calculate the determinant of a matrix?

Yes, there are various other methods to calculate determinant such as row reduction method, eigenvalue method, etc.

Q5. What is the use of calculating determinant of a matrix?

The determinant of a matrix is used to determine various properties of the matrix such as its invertibility, rank, characteristic equation, etc. It finds applications in various fields of science and engineering.

Cara Menghitung Determinan Matriks