1.There are 6 Higher Secondary Schools, 8 High School and 13 Primary Schools in a town. Represent these data in the form of 3 × 1 and 1 × 3 matrices.

Datas are 6,8,13

[6   8  13] → 1 × 3 matrix

2.Find the order of the following matrices.

contains 2 rows and 3 columns

             Therefore order = 2 × 3

  contains 3 rows and 1 column

        Therefore order = 3 × 1

  contains 3 rows and 3 columns

                 Therefore order = 3 × 3

3.A matrix has 8 elements. What are the possible orders it can have?

8 = 8 ×1 and 4 × 2

Therefore possible orders are 8 ×1, 1 × 8, 4 × 2 and 2 × 4

4.A matrix consists of 30 elements. What are the possible orders it can have?

30 = 30 × 1

     = 2 × 15

     = 3 × 10

     = 6 × 5

Therefore possible orders are 30 × 1, 1 × 30, 2 × 15, 15 × 2, 3 × 10, 10 × 3, 5 × 6, 6 × 5

5.Construct a 2 × 2 matrix A = [a_ij] whose elements are given by a_ij = ij

The elements of a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

Here   a₁₁ = 1 × 1 = 1

          a₁₂ = 1 × 2 = 2

          a₂₁ = 2 × 1 = 2

          a₂₂ = 2 × 2 =4

6.Construct a 3 × 2 matrix A = [a_ij] whose elements are given by

The elements of a 3 × 2 matrix are a₁₁, a₁₂, a₂₁, a₂₂, a₃₁, a₃₂.

7.If

(i) Find the order of the matrix

(ii) Write down the elements a₂₄ and a₃₂

(iii) In which row and column does the element 7 occur

i) A has 3 rows and 4 columns.

   Therefore order of A = 3 × 4.

ii) Elements in 2^nd row and 4^th column is 4.

    Elements in 3^rd row and 2^nd column is 0.

iii) 7 is in 2^nd row and 3^rd column.

8. If , then find the transpose of A.

Transpose of A is obtained by interchanging rows and columns of A.

9.If , then verify that (A^T)^T = A

10.Find the values of x, y  and z from the matrix equation

Since the two matrices are equal, corresponding elements are equal.

      5x + 2 = 12,     y - 1 = -8,        4z - 6 = 2

  5x = 10,     y = -7,         4z = 8

  x = 2,         y = -7,           z = 2

Therefore x = 2,  y = -7, z = 2.

11. Solve for x and y if 

Since the matrices are equal,

          2x + y = 5  → (1)

            x - 3y = 13 → (2)

From (1), y = 5 - 2x

Therefore (2) x - 3 (5 - 2x) = 13

                   x - 15 + 6x = 13 7x = 28

                                                   x = 4

Therefore y = 5 -8 = -3

Therefore x = 4 and y = -3

12.If , then find the additive inverse of A

13.Let and . Find the matrix C if C = 2A + B.

14.Find a and b if

2a - b = 10  and 3a +b = 5

   b = 2a - 10

Therefore 3a + 2a - 10 = 5 5a = 15 a = 3

Therefore b = 6 - 10 = -4

Therefore a = 3, b = -4

15.Find the product of (2  -1)

(2  -1)   = 2 × 5 + (-1) × 4

                 = 10 - 4

                 = 6

16.Find the product :

17.If and y AX = C, then find the values of x and y

18.If then show that A² - 4A + 5I₂ = 0

19.If then find AB and BA. Are they equal?

20.If verify that (AB)^T=B^TA^T

21.Prove that are inverses to each other under matrix multiplication.

22.Solve