Exams Prep Master
Determinants and matrices are an integral part of mathematics. They make our calculations far easier when used in the right way. Both of them are calculated to estimate the results of linear equations. Besides, the determinant should be non-zero to find a solution to a set of linear equations. We also use determinants as a function. Likewise, matrices have made their impact in various subjects like statistics, physics, and economics. Here we will get a better understanding of determinants and matrices, their types, properties, and solve some questions.
| Table of Content |
Key Takeaways: Matrix, Determinant, Adjoint, Solution, Algebra, Statistics, Linear equations, Function, Rows, Coloumns
Read more: Logarithms
What is a Matrix?
[Click Here for Sample Questions]
A matrix can be defined as a rectangular table of numbers, symbols, or expressions, which are arranged in the form of rows and columns. In mathematics such a distribution represents the property of an object or linear equations. Suppose, a matrix is of a 3*3 type, it means that there are 3 rows and 3 columns. Or, if a matrix is of 2*3 types, it means there are 2 rows and 3 columns. For example,
This is a 2*4 matrix, it has 2 rows and 4 columns..
It would be wrong to say that matrices are limited only to linear algebra. Matrices are responsible for carrying out a lot of functions and operations in algebra. They also have importance in areas like quantum states, coordinate changes and transformation in physics, electronics, probability, statistics, geometry, and graphs. Many problems are solved in numerical analysis by computing matrices.
The video below explains this:
Matrices Detailed Video Explanation:
Read more: Orthogonal Matrix
Types of Matrix
[Click Here for Sample Questions]
1. Diagonal Matrix
If all the values or entries coming under the main diagonal are 0, the matrix is automatically called an upper triangular matrix. On the other hand, if the values present above the main diagonal are 0, then it is known as a lower triangular matrix.
If all the values present inside the matrix are 0 except the diagonal, it is called a diagonal matrix.
Diagonal matrix A
2. Zero Matrix
A zero matrix is a kind of matrix in which all the entries or values are 0. This is also known as the null matrix in mathematics.
3. Identity matrix
A matrix can be defined as an identity matrix in which all the elements in the main diagonal are equal to 1 whereas the rest of all the elements are 0.
4. Symmetric matrix
When a square matrix is equal to its transpose, it is called a symmetric matrix. For example, given below is a matrix A, and its transpose AT.
In this case, it is called a symmetric matrix.
5. Skew Symmetric Matrix
When matrix A is equal to the negative of its transpose, it is known as a skew-symmetric matrix.
6. Transpose of a Matrix
One can easily find out the transpose of a matrix by interchanging the rows for columns, and the column for rows. It is denoted by the letter T. If a matrix A exists, then its transpose will be written as AT.
For example,
In this case, the original matrix is of a 3 by 4 dimension, whereas, the transpose is 4 by 3 dimension. As we can clearly observe, the rows are made into columns, in order to get the transpose.
7. Inverse of a Matrix
Inverse of a matrix is usually said in relation to a square matrix. An inverse exists for every square matrix. However it should be remembered that inverse will only be possible, when the determinant is non-zero. Suppose, a matrix A exists, its inverse will be A-1. Also, it must satisfy this property,
A*A-1 = (A-1)*A = I
Here, the letter I denotes the identity matrix.
8. Adjoint of a matrix
The adjoint of a matrix is denoted by Adj A. Suppose we have a square matrix [Aij] of order n. The adjoint of such a matrix will be equal to the transpose of the cofactor of matrix A.
Read more: Differential Equations Applications
What is a Determinant?
[Click Here for Sample Questions]
Determinant has been considered as a scalar value. This value serves as the function of all the elements present inside a square matrix. If the matrix is invertible, the determinant is not zero. It is denoted by det(A), or det A, or even |A|. It can be noted that in every matrix of a dimension of 2*2, every determinant is known as a minor of the original matrix.
Given below, it is shown how to find the determinant of a 3*3 matrix. Let us take a look,
Discover about the Chapter video:
Determinants Detailed Video Explanation:
Properties of Determinants
[Click Here for Sample Questions]
- Determinant of a matrix is 0 when all the values inside the matrix are 0
- If an Identity matrix I is present of the order m*n, then the det(I) = 1
- Laplace's formula and the Adjudicate matrix.
- If two square matrices A, and B, have the same size, then det (AB)= det(A).det(B)
- If matrix A-1 is the inverse of matrix A, then det (A-1) = 1/det (A) = det (A)-1
- If matrix AT is the transpose of the matrix A, then det(AT) = det(A)
- Inside a triangular matrix, the determinant is equal to the product of all elements falling in the diagonal.
Also Read:
Things to Remember
- A matrix is a rectangular representation of many elements in the form of rows and columns. These elements may be expressed in the manner of numbers, or symbols. Matrices serve great importance in solving linear equations, graphs, economics and physics.
- There are different types of matrices some of them are null matrix, row and column matrix, diagonal matrix, upper triangular matrix, identity matrix, symmetric and skew symmetric matrix.
- Determinants represent coefficients given in linear equations. They are considered as a scalar quantity. If there exists a matrix A, then its determinant will be written as det(A), or det A, or |A|. If all the elements in a matrix are 0, then the determinant is also 0.
- In the transpose of a matrix, the values of the rows are changed with the values of columns. In the case of square matrices with identical dimension, det AB = det A det B. the determinant in a triangular matrix can be calculated by multiplying all elements in the diagonal.
Read more: Geometric Mean
Sample Questions
Ques. What is a matrix? (2 marks)
Ans. A matrix is a group of numbers that are expressed or distributed in a rectangular formation. Apart from being numbers, they may be certain symbols or expressions present in the form of rows and columns that define a particular object. Apart from algebra, the uses of matrices can be evidently found in various fields.
Ques. Mention some properties of determinants. (2 marks)
Ans. Some common properties of determinants are as follows,
- If all the values inside a matrix are 0, then the determinant will also be 0
- If two square matrices of equal dimensions are present, then, |AB| = |A| |B|
- If a matrix A has a transpose AT, then |AT| = |A|
Ques. Find out the transpose of the given matrix. (2 marks)
Ans. In a given matrix, when we interchange the rows into columns, and the columns into rows, the transpose of the matrix is obtained. It is denoted by the letter T. For example if there is a matrix A, its transpose will be AT.
If A = [ aij ]mn then, A’ = [ aij ]nm.
Ques. What are the different types of matrix? (2 marks)
Ans. In algebra, one can find different types of matrices. These are devised in solving different problems according to the situation. Some of them are,
- Symmetric and skew-symmetric - When a matrix is equal to its transpose, we will call it a symmetric matrix. On the other hand, when a matrix is equal to the negative of its transpose, it is known as a skew-symmetric matrix.
- Null or void matrix - A matrix in which all the elements are 0, is known as a null matrix.
- Identity matrix - An identity matrix is one where all the entries in the main diagonal are 1, whereas the rest of all values are 0.
Ques. Define diagonal matrix. Give an example. (2 marks)
Ans. A matrix in which all the values below as well as above the principal diagonal are 0, it is known as a diagonal matrix. Both lower and upper triangular matrices are present here. A matrix must satisfy two specific conditions in order to be diagonal. These are,
- Must be a square matrix
- if and only if it is triangular and normal
Ques. Is it possible for the determinant to be negative? (2 marks)
Ans. Yes, it is very much possible for the value of the determinant to be negative. As we already know that determinants can be found everywhere in mathematics, and are known to be a real number. This proves that the value of a determinant may be positive or even negative. Fractions are also taken into account. Determinants are not associated with the absolute value.
Ques. What are the applications of matrices? (2 marks)
Ans. Our history has witnessed the importance of matrices since as long as the 2nd century BC. its usage has a wide application in fields like,
- Business organisation - Decision matrices are taken help of while making many business decisions in big firms and organisations. They are well crafted and help prioritise issues and thereby solve them efficiently.
- Mathematics - Matrices help you calculate linear equations. It is very effective in giving results especially when there are very few quantitative parameters, be it in the case of two or three variables.
Ques. What happens when the determinant is 0? (2 marks)
Ans. A determinant becoming 0 is possible when either two rows have the exact same values, or if the columns are identical. When this happens, the volume of the area containing sides of its column or rows is 0. This means that a transformation or change becomes the sole concern, on the basis of which vectors are linearly dependent. This expresses 0 volume.
Ques. What is the inverse of a matrix? (2 marks)
Ans. The inverse of a matrix A is denoted by A-1 .in such a way that if we multiply A with the inverse of A, we will get the identity matrix. This will be identical to matrix A. The formula for inverse of a matrix is
A-1 = (1/|A|)Adj A
Where, |A| is the value of the determinant. It should be noted that a matrix can only have an inverse when the value of the determinant will not be 0.
Ques. Name some uses of matrices. (2 marks)
Ans. Matrices help us in various fields of life. Some of these areas where they are useful are
- Graphs
- Geometry
- Probability and statistics
- Symmetries in physics
- Electronics
Ques. Find the transpose of the given matrix. (3 marks)
[(1, 4, 3); (4, 5, 3); (2, 4, 3)]
Ans. According to the formula,
A = 1 4 3
4 5 3
2 4 3
Then its transpose will be,
1 4 2
A’ = 4 2 3
3 3 3
Ques. Find the product of the given two matrices. (3 marks)
A = [(11, 23); (17, 37)] and B = [(1, 2); (4, 3)]
Ans. The Product of given matrix would be:
Matrix Multiplication ⇒ AB = [(11, 23); (17, 37)]*[(1, 2); (4, 3)]
= [(11+92, 22+69); (17+148, 34+111)] ⇒[(103, 91); (165, 145)]
Ques. If A = [(3, -2); (4, -2)] and I = [(1, 0); (0, 1)] , find k so that A2 = KA - 2I. (3 marks)
Ans. Given that, A2 = kA – 2I ….(1)
To find the value of k, we will substitute the A and I value in (1),
[(3, –2); (4, –2)] [(3, –2); (4, –2)] = K [(3, –2); (4, –2)] –2[(1, 0); (0, 1)],
Now, equate the corresponding values in the matrices, we can find the value of
k = 3k - 2 ⇒ k = 1
-2 = -2k ⇒ k = 1
4 = 4k ⇒ k = 1
-4 = -2k - 2 ⇒ k= 1
So, all the k values are equal to 1. Therefore, k = 1.
Ques. Find the values of x and y if [(3x+7, 5); (y+1, 2-3x)] = [(0, y-2); (8, 4)] (3 marks)
Ans. Since the given equations are equal, we can equate the corresponding elements. Thus, we get: 3x + 7 =0 ⇒ x = -7/3
y-2 = 5 ⇒ y = 7
y + 1 = 8 ⇒ y = 7
2-3x = 4 ⇒ x= -2/3
As x and y cannot have two values, there is no solution for the given set of equations.
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments