Matrices: Types, Properties, Formulas & Examples

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Matrices is defined as the arrangement of numbers, expressions or symbols formed in a rectangular shape. It is organized in the form of rows and columns, which are arranged in horizontal and vertical forms. The total size of the matrix is determined by the number of rows and columns. 

  • Matrices is a method by which we can easily solve some complex expressions.
  • It helps us study a circuit's charged resources with voltage, resistance, amper, etc. 
  • The concept helps in projecting, operating and studying linear graphs.
  • Matrices are also useful in representing quadratic forms. 
  • It calculates the coefficient of expressions or equations in a meaningful way.
  • Matrices are used in many areas like Magnification, Cost Estimation or in the sales projection.

Read More: Matrix Addition

Key Terms: Matrices, Column Matrices, Row Matrices, Square Matrices, Diagonal Matrices, Zero Matrices, Scalar Matrices, Identity Matrices


What are Matrices?

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Matrices is defined as a rectangular arrangement of m and n elements. The matrices arrangement consists of a m number of rows and n number of columns enclosed within the square brackets. 

  • Matrices are shown within the [] or () clauses.
  • It is used in cryptographical techniques.
  • The entries in a matrix are called elements.
  • Horizontal entries of a matrices are called rows.
  • Vertical entries of a matrices are called columns.
  • Different operations like addition and subtraction can easily be performed on matrices.

Read More: Upper Triangular Matrix

Solved Example of a Matrices

Suppose, If we have to say that Kiran has 20 Books and 30 Pens then we can denote it as [20, 30] where 20 shows the number of Books and 30 Shows the number of Pens Kiran has. Now, an another person named Tushar is also standing along with the Kiran and He says that He has 25 Books and 12 Pens then If we have to convert that information in the meaningful way then we can write it as :

Books Pens

Kiran 20 30

Tushar 25 12

Here in the above table [20, 30] and [25, 12] are considered as rows of Matrices and [20, 25] and [30, 12] are considered as the columns of the Matrices. Above Shown Matrix is called as a 2*2 matrices where Number of Rows = Number of Columns = 2. 

Read More: Orthogonal Matrix

Types Of Matrix

Types Of Matrix

The video below explains this:

Matrices Detailed Video Explanation:


Types Of Matrices

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There are many types of matrices which can be used in mathematical calculation. Here are the list of different types of matrices.

Column Matrix 

A matrix with only one column is called a Column Matrix. In general, we can say that in the column matrix number of rows is equal to zero but number of columns is equal to one. 

Row Matrix 

A matrix having only one row is called a row natrix. In general, we can say that in the row matrix number of columns is equal to zero but number of rows is equal to one.

Read More: Matrix Multiplication

Square Matrix 

A matrix having same number of rows and same number of columns is called a Square Matrix. If the size of a Matrix is m*n then in the Square Matrix there is m=n .

Diagonal Matrix 

A matrix that has an element only on diagonal position then it is called a Diagonal Matrix. 

Zero Matrix 

A matrix having a zero on all the positions is called a Zero Matrix.

Scalar Matrix 

A diagonal matrix having the same elements on diagonal Position is called as a Scalar Matrix.

Identity Matrix 

In the square matrix, in which all elements on diagonal positions are equal to one and rest elements are 0 called an Identity Matrix.

Solved Example of Type of Matrices

Example 1:

Matrix A 

1  0  0

0  1  0

0  0  1

Here as shown in above example the matrix is of 3*3 size and 1 is written on only diagonal positions. So, We can say that it is a diagonal matrix.

Read More:


Operation on Matrices 

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The list of operation performed on matrices are as follows:

Addition 

Addition of two matrices means we have to add the elements of one matrix with the other matrix in row wise or column wise. It means we have to do addition of elements. 

Multiplication 

Multiplication of matrices means we have to multiply the elemnets of one matrix with the other matrix in row wise or column wise. It means we have to do multiplication of elements. 

Subtraction 

Subtraction of matrices means we have to subtract the elemnets of one matrix with the other matrix in row wise or column wise. It means we have to do subtraction  of elements. 

Read More: Applications of Determinants and Matrices

Solved Examples of Operation of Matrices

Given below are the examples of operation of matrices

Example 1 : Addition of two matrices

Matrix A

10  20  30

40  50  60

70  80  90

Matrix B

36  25  65

23  25  63

41  25  33

Now If we want to add these two matrices then we have to do addition by elements i.e first we have to add the element on 1st row and 1st column and result will be placed at 1st position of resultant matrix.Then Same thing is done with the remaining elements and their positions.

Resultant Matrix A+B

46    45     95

63    75   123

111 105   123

Example 2: Subtraction of two matrices

Matrix A

2  6  3

4  9  3

3  8  7

Matrix B 

3  1  6 

7  1  0

1  3  2

Now If we do the multiplication of two matrices A and B then the resultant matrix is shown below.

Resultant Matrix C

6   6  18

28 9   0

3  24 14

Read More: Types of Matrices

Example 3: Multiplication of two matrices

Matrix A

2  6  3

4  9  3. 

3  8  7

Matrix B

3  1  6 

7  1  0

1  3  2

Resultant Matrix C

-1  5  -3

-3  8   3

 2  5   5

Read More: Inverse Matrix Formula


Properties of Matrices 

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The important properties of matrices are as follows:

  • -A = (-1)A
  • A – B = A + (-1) x B
  • A + B = B + A
  • (A + B) + C = A + (B + C), where A,B and C are in the same order of matrix.
  • k(A + B) = k x (A) + k x (B), where k is a constant and A,B are in same order of matrix
  • ((A)’)’ = A
  • (A + B)’ = A’ + B’
  • (AB)’ = B’A’

Read More:


Important Formulas of Matrices

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The important formulas of matrices are as follows:

  • A-1 = adj(A)/|A|
  • |adj A| = |A|n-1 where n is the order of matrix A
  • A(adj A) = (adj A)A = I, where I is an Identity Matrix
  • adj(adj A) = |A|n-2A where n is the order of the matrix
  • adj(AB) = (adj B)(adj A)
  • |adj(adj A)| = |A|(n-1)^2
  • adj(Ap) = (adj A)p
  • If A is symmetric then adj(A) is also symmetric
  • adj(kA) = kn-1(adj A) where k is any real number
  • adj(I) = I
  • If A is a diagonal Matrix then adj(A) is also a diagonal matrix
  • adj 0 = 0
  • If A is a triangular matrix then adj(A) is also a triangular matrix
  • If A is a singular Matrix then |adj A| = 0

Read More: Operations on Matrices


Things to Remember

  • Matrices is a rectangular table of numbers, symbols or expressions.
  • The elements are arranged in the form of rows and columns.
  • It is used to represent linear map and can solve problems on linear algebra.
  • Addition, subtraction and multiplication are some of the operations performed on matrices.
  • The size of a particular matrix can be defined by the number of rows and columns.

Read More: Transpose of a Matrix


Previous Years Questions


Sample Questions

Ques. What is the Use of Matrices ? (2 marks)

Ans. Matrices is defined as a rectangular arrangement of m and n elements. The matrices arrangement consists of a m number of rows and n number of columns enclosed within the square brackets. Matrices are shown within the [] or () clauses. It is used in cryptographical techniques. It used in many areas like Magnification, Cost Estimation or in the sales projection etc. 

Ques. Can we apply the Inverse Operation on Matrices ? (2 marks)

Ans. Yes, We can apply the Inverse Operation on the matrix. By applying Inverse operation, rows are converted to columns and columns are converted to the rows.The inverse of the matrix can be find out using the formula A-1 = (1/|A|)(adj A)

Ques. Give some names of algorithms which use a matrix in their method? (5 marks)

Ans. Algorithms like Floyd-Warshall, Travelling Salesman algorithm use a matrix in their method.

Algorithms like Floyd-Warshall, Travelling Salesman algorithm use a matrix in their method. 

Algorithms like Floyd-Warshall, Travelling Salesman algorithm use a matrix in their method. 

Algorithms like Floyd-Warshall, Travelling Salesman algorithm use a matrix in their method. 

Algorithms like Floyd-Warshall, Travelling Salesman algorithm use a matrix in their method. 

Check-Out: 

CBSE CLASS XII Related Questions

  • 1.
    For a function $f(x)$, which of the following holds true?

      • $\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$
      • $\int_a^b f(x) dx = 0$, if $f$ is an even function
      • $\int_a^b f(x) dx = 2 \int_0^a f(x) dx$, if $f$ is an odd function
      • $\int_0^a f(x) dx = \int_0^a f(2a + x) dx$

    • 2.
      Evaluate \( \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \)


        • 3.

          Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
          Let \( A_1 \): People with good health,
          \( A_2 \): People with average health,
          and \( A_3 \): People with poor health.
          During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.
          Based upon the above information, answer the following questions:
          (i) A person was tested randomly. What is the probability that he/she has contracted the disease?}
          (ii) Given that the person has not contracted the disease, what is the probability that the person is from category \( A_2 \)?


            • 4.
              Let \[ A = \begin{pmatrix} 1 & 4 \\ -2 & 1 \end{pmatrix} \quad \text{and} \quad C = \begin{pmatrix} 3 & 4 & 2 \\ 12 & 16 & 8 \\ -6 & -8 & -4 \end{pmatrix}. \] Then, find the matrix $B$ if $AB = C$.


                • 5.
                  If \( f(x) = \begin{cases} \frac{\sin^2 ax}{x^2}, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of 'a' is :

                    • 1
                    • -1
                    • 0
                    • \( \pm 1 \)

                  • 6.
                    If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

                      CBSE CLASS XII Previous Year Papers

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