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The determinant is considered a scalar associated with a square matrix. The association of a determinant is applicable for every square matrix, and determinants are exclusively defined for square matrices only. Therefore, understanding the basic properties of Determinants are very crucial for the analysis and solving of linear equation systems.
Determinants are scalar quantities and can be procured by adding the products of the entries of a square matrix and its cofactors depending on a designated rule. The cross result of two vectors can be solved effortlessly via the calculation of determinants. In simpler words, a determinant can be described as a function with an input of a square matrix, and the output is a number.
Basic concept on Determinants.
Determinants can be considered as a scaling element of matrices. It takes a square matrix as the input and solves to give a single number as the output.
For example: if we consider n to be the number of rows and columns of a matrix, we can say it is a n x n matrix. The simplest form of a square matrix can be designated as 1 x 1 matrix.
If we dig into some complex forms of determinants, we have to understand the basic properties of Determinants of the 2 x 2 matrix.
Integral properties of determinants.
For effortlessly working with determinants, it is extremely important to know about the basic properties of Determinants. The overall work related to determinants is not very difficult if the concept about the basic properties of Determinants is clear, and with rigorous practice, it becomes easier.
There are a total of eleven properties associated with determinant solving. They are of the following types.
1.Reflective property.
According to this property, the determinant remains the same even if the rows and columns are interchanged.
Example: =
a d g | b e h | c F i |
, so we can say that the determinant of the particular matrix and the transpose of the matrix will be the same.
2.Property of proportionality.
According to this property, if two rows or columns in a matrix are the same, we can say that the determinant of the matrix will be zero.
3.Switching property.
According to this property, if we interchange any two rows or columns in a matrix, then the sign of the determinant will also change, keeping the absolute value the same.
4.Scalar property.
According to this property, if we multiply all the elements of a row or column in a matrix by any real number, then the resulting value of the determinant of that matrix will also be multiplied by the same real number.
5.Summation property.
According to this property, if all the entries of the determinant are shown as the sum of two or more entities, then the determinant will be expressed as the sum of the two or more determinants.
Example:
a+1 d g | b+1 e h | c+1 f i |
I=
a d g | b e h | c F i |
+
1 d g | 2 e h | 3 F i |
1.Invariance property.
Suppose any scalar multiples of corresponding entries of other rows or columns are added to every entry of any row or column of a determinant. The determinant will stay unchanged even if we apply any operation to it.
I=
a d g | b e h | c F i |
=
a+k d g | b+q e h | c+r f i |
Property of singularity.
According to this property, if the determinant of a matrix is zero, then the matrix itself is said to be singular.
1.Factor property.
If we take c=k, and by putting this value if the determinant becomes zero, in such a case, we can conclude that (c-k) is considered to be the factor of the determinant.
2.Cofactor property.
The determinant of any matrix is the same when compared to the determinant of the cofactors of all the elements in the matrix.
3.Multiplication property.
The multiplication product of two matrices is equivalent to the product of the individual determinants of the matrices.
det(CD)=det C X det D.
For calculation of area of a triangle.
- While understanding the basic properties of Determinants, it is important to understand its aspect in calculating the area of a triangle.
- With the property of determinants, one can calculate the area of a triangle with the following information.
- If the vertices of a triangular are as (x1, y1), (x2, y2) and (x3, y3), then the area of the triangle can be shown as:
∆= ½
x1 x2 x3 | y1 y2 y3 | 1 1 1 |
Some solved problems:
- Find the value of the determinant
6 9 | 3 4 |
Answer: Given
6 9 | 3 4 |
= 6 x 4 -3 x 9 = 24 – 27 = -3.
- Show that, det A= det Aᵀ, for the matrix A=
1 2 | 3 4 |
Answer: given A=
1 2 | 3 4 |
Determinant of the matrix, det A =
= 4 x 1 – 3 x 2 = 4-6= -2(1)
We know that if the rows and columns are interchanged in a matrix, we can obtain the transpose of the matrix.
Hence, Aᵀ=
1 3 | 2 3 |
det Aᵀ= 4 x 1 – 2 x 3= 4-6= -2(1)
Hence, we can say det A= det Aᵀ.
Conclusion
While working with determinants, understanding the basic properties of Determinants is extremely important. The properties of determinants will help get an idea about how to solve a sum. With proper practice and a clear understanding of determinants’ basic properties, solving the sums will become effortless.
A few of the fundamental aspects like the all-zero property, property of summation, switching property, calculation of area of triangles should be studied thoroughly to solve the problems. Other than that, study the aspects of the 2 x 2 and 3 x 3 matrix to get a clear idea.
