1. 1 Lecture May 2011 Matrices and Determinants Mathematics XI 2011

2. 2 1.1Matrices 1.2Operations of matrices 1.3Types of matrices 1.4Properties of matrices 1.5Determinants 1.6 1.7 Inverse of a 3  3 matrix Application

3. 3 Zero matrices  Every element of a matrix is zero, it is called a zero matrix, i.e., 1.1 Matrices

4. 4 Sums of matrices 1.2 Operations of matrices  Two matrices of the same order are said to be conformable for addition or subtraction.  Two matrices of different orders cannot be added or subtracted, e.g., are NOT conformable for addition or subtraction.

5. 5 1.3 Types of matrices  Identity matrix  The inverse of a matrix  The transpose of a matrix  Symmetric matrix  Orthogonal matrix

6. 6  (AB) -1 = B -1 A -1  (A T ) T = A and ( A) T =  A T  (A + B) T = A T + B T  (AB) T = B T A T 1.4 Properties of matrix

7. 7 1.5 Determinants Consider a 2  2 matrix: Determinant of order 2  Determinant of A, denoted, is a number and can be evaluated by

8. 8 1.6 Inverse of a 3  3 matrix Cofactor matrix of The cofactor for each element of matrix A:

9. 9 Cofactor matrix of is then given by: 1.6 Inverse of a 3  3 matrix

10. 10 1.6 Inverse of a 3  3 matrix Inverse matrix of is given by:

11. 1.7 Applications Cement Steel Floor(sq.ft) Banglow 200 Yds 300 2500 2000 Banglow 300 Yds 500 3500 3000 Banglow 500 Yds 750 5000 5500 Khi Lhr Isl 450 375 520 330 350 410 120 90 150 Total Cost Khi Lhr Isl Banglow 200 Yds 1200000 1167500 1481000 Banglow 300 Yds 1740000 1682500 2145000 Banglow 500 Yds 2647500 2526250 3265000 11

12. Students can also create parallel examples in Financial institution Industries Time management 12

13. By : Syed Shujaat Hussain Special Thanks to My Master Trainer Mr. Adnan Kiyani 13