## Determinants in Discrete mathematicsDeterminants can be described as the scalar quantities which will be obtained when we add the products of elements of a square matrix and their cofactors on the basis of a prescribed rule. With the help of determinant, we will be able to determine the adjoint, inverse of the matrix, and many more. In case of the determinant, horizontal lines are used to indicate the rows, and vertical lines are used to indicate the columns. If there is a determinant with n order, in this case, it will have n rows and m columns.
The determinant of this matrix is described as follows: In this example, we have two determinants in which the first determinant contains the 2 order and the second determinant contains the 3 order. If there is a square matrix A with an order n*n, in this case, there must be a number associated with it which will be known as the determinant of a square matrix. 1. If there is a matrix with an order of 1*1, then the determinant of it will be A = [a]. 2. Suppose there is a matrix with an order 2*2 The determinant of this 2*2 matrix will be 3. Suppose there is a matrix with an order 3*3 The determinant of this matrix will be ## Note: If there is a determinant with an order n, then the number of elements will be n |

|D| = (2)(5) - (4)(1) |D| = 10 - 4 = 6 | |D| = 2* {(1)(5) - (2)(1)} |D| = 2* (5-2) =2*3 = 6 |

In both matrices, we can see that the determinant remains the same.

There are some other important properties of determinant, which is described as follows:

- If there is a square matrix, then it will be known as invertible iff det(C) ≠ 0.
- If there are two square matrices, A and B which contains the order n*n, in this case, det(BC) = det(B) * det(C) = det(C) * det(B).
- We can indicate the relationship between a matrix D's determinant and its adjoint in the following way: D * adj(D) = adj(D) * D = |D| * I. Here I is used to indicate the identity matrix, and D is used to indicate the square matrix.

If we try to perform the rows and columns operations, then there are some rules which will be useful. These rules are described as follows:

- If we change the rows into columns and columns into rows, in this case, the value of determinant will not be changed.
- If we interchange any two rows or two columns, in this case, the sign of determinant will not be changed.
- The value of determinant of the given matrix will be 0 if and only if any two rows or columns are the same.
- If there is a constant in the matrix that is multiplied by every element of a particular row or column of that matrix, in this case, the constant will also be multiplied by the value of determinant.
- If each and every element of a row or column is expressed in the form of a sum of elements, in this case, the determinant will also be expressed in the form of a sum of determinants.
- If the value of determinant will not be changed if we add or subtract the elements of a row or column with corresponding multiples of elements of another row or column.

There are some important points that we should keep in mind while learning the concept of determinants. These points are described as follows:

- We can consider determinant as a function. That means it can take a square matrix in the form of input and return a single number in the form of output.
- A matrix will be known as a square matrix if it contains the same number of rows and columns.
- If there is a simplest square matrix with an order 1*1, then the determinant of this matrix will be the number itself because the 1*1 matrix contains only one number.

There are a lot of examples of determinants, and some of them are described as follows:

**Example 1:** In this example, we have a matrix A, and we have to determine the determinant of this matrix. The elements of this matrix are described as follows:

**Solution:** The determinant of the above matrix can be written in the following way:

Now we will determine the determinant of it by multiplying the diagonals and subtracting the products in the following way:

|A| = (4*2) - (3*1)

= 8 - 3

= 5

**Answer: det(A) or |A| = 5**

**Example 2:** In this example, we have a matrix C with 2*2 order, and we have to determine the determinant of this matrix. The elements of this matrix are described as follows:

**Solution:** The determinant of the above 2*2 matrix can be written in the following way:

Now we will determine the determinant of it in the following way:

|C| = {(4)(4) - (8)(2)}

= 16-16

= 0

**Answer: The determinant of matrix C is = 0.**

**Example 3:** In this example, we have a matrix A with 3*3 order, and we have to determine the determinant of this matrix. The elements of this matrix are described as follows:

**Solution:** Here, we will use the first row and expand the determinant of above 3*3 matrix.

= 1 * (-1 - (-9) - 3 * (-3 - (-6) + 2 * (-9 - (-2))

= 1* (-1 + 9) - 3* (-3 + 6) + 2 * (-9 + 2)

= 8 - 9 - 14

= -15

**Answer: The determinant of matrix A is -15.**

**Example 4:** In this example, we have to determine the value of determinant with the help of properties of determinants. The determinant is described as follows:

**Solution:** With the help of one of the properties of determinant, we will perform the elementary row transformation R_{1} → R_{1} + R_{2} + R_{3} on the above determinant. According to this property, the value of determinants will not be changed while doing the elementary row transformation. After performing the elementary row transformation (R_{1} → R_{1} + R_{2} + R_{3}), the determinant will become the following:

Now we will use one more property of determinant that says if the row or column of a matrix is 0, then the determinant of this matrix will also be 0. In the above matrix, the first row is 0. That's why the value of above determinant is 0.

**Answer: The value of determinant is 0.**

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