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Mathematics. Matrices and Determinants-1 Session.

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Presentation on theme: "Mathematics. Matrices and Determinants-1 Session."— Presentation transcript:

1 Mathematics

2 Matrices and Determinants-1 Session

3 Matrix Types of Matrices Operations on Matrices Transpose of a Matrix Symmetric and Skew-symmetric Matrix Class Exercise Session Objectives

4 Matrix A matrix is a rectangular array of numbers, real or complex. Column Row Element of m th row and j th column

5 Order of a Matrix A matrix with m rows and n columns has an order m x n. Examples:

6 Example - 1 A matrix has 16 elements, what is the possible number of columns it can have. The possible orders for the matrix are (1 x 16), (2 x 8), (4 x 4), (8 x 2),(16 x 1) So, the number of possible columns are 16, 8, 4, 2 and 1. Solution :

7 Here i can take the values 1 and 2 and j can take the values 1, 2 and 3. Hence, the order of the matrix is (2 x 3). Example-2 Solution : Now, Write the matrix given by the rule Hence, the matrix is

8 Row matrix: Column matrix: Types of Matrices

9 Zero matrix : Square matrix: Diagonal matrix: Types of Matrices

10 Scalar matrix: Identity matrix: Types of Matrices

11 Two matrices A = [a ij ] and B = [b ij ] are equal, if they have the same order and a ij = b ij for all i and j. Equality of Matrices Example:

12 Addition of Matrices If A= [a ij ] and B= [b ij ] are two matrices of the same order, then their sum A + B is a matrix whose (i, j) th element is Example:

13 Multiplication of a Matrix by a Scalar Example: is a matrix and k is a scalar, then

14 Properties of Addition If the order of the matrices A, B and C is same, then (i) A + B = B + A (Commutativity) (ii) (A + B) + C = A + (B + C) (Associativity) (iii) If m and n are scalars, then (a) m(A + B) = mA + mB (b) (m + n)A = mA + nA

15 Find X, if Y= and 2X+Y = Y= and 2X+Y = Example - 3 Solution : 2X + =

16 Find a matrix C such that A+B+C is a zero matrix, where A= and B = Example - 4 Solution : A + B + C = 0

17 Let A= [a ij ] m x n be a m x n matrix and B = [b ij ] n x p be a n x p matrix, i.e., the number of columns of A is equal to the number of rows of B. Then their product AB is of order m x p and is given as Multiplication of Matrices

18 Example If A= and B = then AB is given as

19 If both sides are defined, then (i) A(BC) = (AB)C (Associativity) (ii) A ( B + C ) = AB + AC and (A + B) C = AC + BC ( Multiplication is distributive over addition) Properties of Multiplication of Matrices

20 Example - 5 Solution :

21 Example - 6 If A=, then show that

22 Solution : Example - 7 If A = and I=, then find k if

23 Comparing the corresponding elements of the two matrices, we get 3k-2 = 1, -2k = -2, 4 = 4k, -4 = -2k –2 Taking any of the four equations, we get k=1 Solution Contd.

24 Show that A = satisfies the equation A 2 – 12A + I = O. Example - 8 Solution : Hence, A 2 – 12A + I=O

25 A matrix obtained by changing rows into columns or columns into rows is called transpose of the matrix ( say A ). If the matrix is A, then its transpose is denoted as A T or A ’. Transpose of a Matrix For Example: Consider the matrix The transpose of the above matrix is

26 Example - 9 then verify that (A+B) T =A T +B T Solution: Hence, (A+B) T =A T +B T

27 If A = and B = find Example - 10 Solution :

28 Find the values of x, y, z if the matrix obeys the law AA ’ = I. Example - 11

29 Equating the elements of column 2, we get Solution (Cont.) 2y 2 – z 2 = 0 …(i) Adding (ii) and (iii), we get Form (i), z 2 = 2y 2

30 Solution (Cont.) Putting the value of x 2 and z 3 in (ii), we get Putting the value of y 2 in (i), we get

31 Solution : Example - 12

32 A square matrix A is called a symmetric matrix, if A T = A. A square matrix A is called a skew- symmetric matrix, if A T = - A. Any square matrix can be expressed as the sum of a symmetric and a skew- symmetric matrix. Symmetric and Skew – Symmetric Matrix

33 Show that A= is a skew-symmetric matrix. Solution : Example - 13 As A T = - A, A is a skew – symmetric matrix

34 Express the matrix as the sum of a symmetric and a skew- symmetric matrix. Solution : Example - 14

35 Solution Cont.

36 Therefore, P is symmetric and Q is skew- symmetric. Further, P+Q = A Hence, A can be expressed as the sum of a symmetric and a skew -symmetric matrix. Solution Cont.


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