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Determinant is a number that is computed for a square matrix. A determinant is used in linear equations and in calculations of matrices to write them in a simpler form, and provide an easy way of solving problems. Here, we will discuss the determinant formula, properties, examples, and applications.
Also Read: Applications of Determinants and Matrices
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Key Terms: Determinants, Matrix, Inverse of a Matrix, Algebra, Scalar, Linear Equations, Transpose, Inverse, Matrices, square matrix, special number
Determinant
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Determinant corresponds to a scalar value and it is computed by using the elements of a square matrix. The main difference between a matrix and a determinant is that a matrix is an array of numbers, and a determinant is the special number or value of a square matrix.It is calculated by considering the elements in the top row and the corresponding minor elements of a matrix. Matrices that can be used for determinants have to follow some rules or properties of transformations. Suppose a square matrix is denoted by an alphabet ‘A’, then the determinant of a square matrix A will be denoted by |A|.
Determinant Formula
To calculate a determinant, the very first element of the top row is taken and multiplied by the corresponding minor. This is then subtracted with the product of the second element and its corresponding minor. This is continued till all the elements and their corresponding minors are considered.
Discover about the Chapter video:
Determinants Detailed Video Explanation:
Also Read: Minors and Cofactors of determinants
Determinant Formula
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The determinant formula is helpful to get the determinant of a given square matrix. The determinant formula is used only for square matrices. Basically, when a matrix has an equal number of rows and columns, then we call the matrix a square matrix. For easy calculation purposes, you can easily change a square matrix to a determinant and then use the determinant formula. Let us look at the determinant formula for some basic square matrices.
- 2x2 Matrix Determinant Formula
So, the determinant formula for this is:
Determinant |A| = ad - bc
-
3x3 Matrix Determinant Formula
Let us compute the determinant of a 3 by 3 matrix with an example as follows:
The determinant formula for the 3 by 3 matrix is to be calculated a little more carefully. There are complications of signs (plus or minus). The numbers 1,2,3,.... act as variables and also represent their position numbers.
The determinant formula for the 3x3 matrix consists of three terms. We need to start with positive signs for the first term, then negative signs for the second term, and lastly, again positive signs for the last term. Thus, the signs have to be assigned alternatively like plus, minus, and plus.
Therefore the formula is,
Determinant |A|= 1(5x9 - 8x6) - 2(4x9 - 7x6) + 3(4x8 - 7x5)
= 0
Also Read: Types of Matrices
Properties of Determinants
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Listed below are the important properties of determinants:
- After interchanging any two rows of a determinant with each other, the sign of the determinant is changed. The same rule goes with columns as well.
- If the rows and columns are interchanged, then the value of the determinant will not get affected at all. It will stay the same.
- In case, any two rows or columns of a determinant are identicals, then it is sure that the value of the determinant will be only zero.
- If ‘k’ is a constant, and then we multiply ‘k’ with any one of the rows or columns in the determinant.The value of the determinant will also become ‘k’ times it was before.
- In a triangular matrix, the determinant is equal to the product of the diagonal elements.
- The determinant of a matrix is zero if all the elements of the matrix are zero.
Some of the other important properties are summarised in the table below:
| Condition | Property |
|---|---|
| Multiplicative Property of Determinant | |AB| = |A| X |B| |
| If An is an identity matrix, then | |A| = 1 |
| If the MT is the transpose of matrix M, then | |MT| = |M| |
| If M-1 is the inverse of matrix M, then | |M-1|= |M-1| |
| If two square matrices M and N have the same size, then | |MN|= |M| X |N| |
| If matrix M has a size axa and C is a constant, then | |CM| = Ca X |M| |
| If X, Y, and Z are three positive semidefinite matrices of equal size, then | |X+Y| ≥ |X| + |Y| for X,Y, Z ≥ 0 |X+Y+Z| + |C| ≥ |X+Y|+ |Y+Z| |
Also Read: Properties of Determinants
Important ExampleExample: Verify the property that the determinant value remains unchanged if the rows and columns are interchanged for the following Matrix A.
Solution: Expanding the matrix A to determine the determinant, we get |A| = 2(0-20) + 3(-42-4) + 5(30-0) =2 (-20) + 3(-46) + 5(30) = -40=138+150 = -28 On interchanging the rows and columns, the matrix A becomes, A1
Expanding A1 to evaluate |A1|, we get |A1| = 2(0-20) + -6(21-25) + 1(-12+0) = 2(-20) + [(-6)(-4)] +1(-12) =-40+24 -12 = -28 Hence, |A| = |A1| The property is hence proved. |
Also Read: Order of a Matrix
Applications of Determinant Formula
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The determinant formula has wide applications in the field of science and maths. Some of them are listed below:
- Determinants are used to check the consistency of a given system of linear equations having one or more variables.
- Consistency can be checked if the equations satisfy the following two conditions:
- The system of equations has one or more solutions, then it is consistent.
- The system of equations does not have a solution, then it is inconsistent.
- They can be applied vastly in engineering, science, social science, and economics.
- Determinants help to determine the inverse of a given matrix.
- In geometry, determinants are used to find the area of a triangle if the vertices are known.
Also Read: Singular Matrix
Things To Remember
- Determinant is single value evaluated with the help of the elements and thier corresponding cofactors.
- The determinant can be evaluated for square matrices only.
- The determinant of a 2 by 2 matrix is Determinant = ad - bc, where a, b, c, and d are the elements of the square matrix.
- While solving the determinants, we must make sure tto use the rules of positive and negative signs that go go alternatively.
- Determinants have a wide level of applications in various fields like science and engineering.
- In linear algebra, determinants are used to find the consistency of a given equation.
- For a matrix A made by interchanging the positions of two rows or columns of another matrix B, then |A| = -|B|.
Also Read:
Sample Questions
Ques 1. Below is a given matrix: (2 marks)
Find the value of its determinant.
Ans. The Determinant of the required matrix will be:
Det = 2×10 – 6×3
= 20 – 18
= 2
Ques 2. Below is the given matrix of order 3 x 3: (2 marks)
Using the determinant formula, find the determinant value of the matrix.
Ans. Applying the determinant formula on the given matrix, we will get-
= 5×(2×4 – 5×6) - 8×(4×4 – 8×6) + 7×(4×5 – 8×2)
Ques 3. If 
, then find the value of ‘x’. (2 marks)
Ans. We have been given that these two 2x2 matrices are equal. It means the corresponding
elements of the matrices must be equal. So, we have:
3x = 15
x = 15/3
x = 5
Ques 4. Find the adj A for 
. (2 marks)
Ans. We have to find the adj A, in the 2x2 order matrix, it is not so complicated.
You can just interchange the place of 8 and 6 with each other, and then change the sign
of the 1 and 5.
We will get-
Ques 5. Find the determinant value of: (3 marks)
Ans. We would just expand the given determinant along the first row.
After expanding along the first row, we will get-
Det = cosα cosβ x (cosβ cosα – 0) – cosα sinβ x (–sinβ cosα) – sinα x (–sinα sin2β – sinα cos2β)
= cos2α cos2β + cos2α sin2β + sin2α sin2β+ sin2α cos2β
= cos2α x (cos2β + sin2β) + sin2α x (sin2β + cos2β)
= cos2α· 1 + sin2α· 1
= 1
Ques 6. Find the determinant of the square matrix that is given below: (3 marks)
Ans: We have been given is-
a = 1; b = 0; c = 4
d = 4; e = 5; f = 7
g = 2; h = -3; i = 9
Using determinant formula, we will get-
Det = a(ei - fh) - b(di - fg) + c(dh - eg)
= 1(5×9 + 3×7) − 0(-2×9 − 2×7) + 4(-2×-3 − 2×5)
= 1(54 + 21) − 0 + 4(6 - 10)
= 75 - 16
= 59
Ques 7. What will be the determinant value of: (2 mark)
Ans. As we have clearly mentioned in the article that if any row or column is filled with
zeros, then the determinant value of the matrix will also become zero.
Ques 8. If A’ is the transpose of a square matrix A, then: (2 mark)
|A| + |A’| = 0
|A| = |A’|
|A| ≠≠ |A’|
None of these
Ans. The correct answer is option ‘B’.
|A| = |A’|
It is one of the properties of transpose.
Previous Year Questions
- Find the solution of the equation by using the determinant formula. [BITSAT 2006]
- Evaluate the determinant of the two matrices A and t. [BITSAT 2006]
- Apply determinant properties to solve the system of 3 equations. [AP EAPCET]
- Find the area of a triangle with the given vertices. [KCET 2017]
- Find the constant term in the polynomial by applying property of determinant. [KCET 2010]
- If two matrices are in A.P and find the range of the determinant. [CBSE Clas XII]
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