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Mathematics

Mathematics. Session. Matrices and Determinants-1. Session Objectives. Matrix Types of Matrices Operations on Matrices Transpose of a Matrix Symmetric and Skew-symmetric Matrix Class Exercise. Column. Row. Element of m th row and j th column. Matrix.

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Mathematics

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  1. Mathematics

  2. Session Matrices and Determinants-1

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

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

  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. Solution : 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.

  7. Write the matrix given by the rule Hence, the matrix is Example-2 Solution : 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). Now,

  8. Row matrix: Column matrix: Types of Matrices

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

  10. Types of Matrices Scalar matrix: Identity matrix:

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

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

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

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

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

  17. Multiplication of Matrices Let A= [aij]m x n be a m x n matrix and B = [bij]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

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

  19. Properties of Multiplication of Matrices 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)

  20. Example - 5 Solution :

  21. If A= , then show that Example - 6 Solution :

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

  23. Solution Contd. 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

  24. Show that A = satisfies the equation A2 – 12A + I = O. Example - 8 Solution : Hence , A2 – 12A + I=O

  25. For Example: Consider the matrix The transpose of the above matrix is Transpose of a Matrix 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 AT or A’ .

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

  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 Solution :

  29. Solution (Cont.) Equating the elements of column 2 , we get 2y2 – z2 = 0 …(i) Adding (ii) and (iii), we get Form (i), z2 = 2y2

  30. Solution (Cont.) Putting the value of x2 and z3 in (ii), we get Putting the value of y2 in (i), we get

  31. Example - 12 Solution :

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

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

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

  35. Solution Cont.

  36. Solution Cont. 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.

  37. THANK YOU

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