1. 1. Contents History of matrices Introduction of matrices Definition of matrix Matrix Multiplication Row & column Order of matrix Compact form of matrix Determinants of ad joint inverse of a matrix Ad joint Matrix Inverse Matrix Inverse Matrix Method Gauss-Jordan's Metod Steps to convert the augmented matrix into fins=al matrix & get solutions
2. 2. James Joseph Sylvester: He was British mathematician. He was born in London ,England. His father Abraham joseph was a merchant. James adopted the summaries Sylvester when his older brother did so uponemringtion united states at age of 14. Sylvester was a student of Augustus de Morgan at the university of London. Arthur Clayey: He was British mathematician who helped founding the British school of pure mathematics. Although clayey was born in England, his first seven years were spent in St. Petersburg Russian where his parents lived in the family permanent return to England in 1828. He was educated at a small private school in blackheart followed by the three - years course at king college, London.
3. 3. Matrices Definition of MATRIX: A matrix in general sense, represents a collection of information stored or arranged in an orderly fashion. Application of Matrices: There are numerous application of matrices, both in mathematics and other sciences. Some of them, game theory and economics, text mining and automated thesaurus compilation computer graph theory, network theory makes use of matrices in various way.
4. 4. Matrix Multiplication Scalar Multiplication Consider the following If s is a scalar and A is a matrix 1 5 s = 2, and A = 4 3 2 1 1 5 2(1) 2(5) 210 sA = 2 * 4 3 = 2(4) 2(3) = 8 6 2 1 2(2) 2(1) 4 2 Rule: If s is a scalar and A is a matrix, then the scalar multiple sA is the matrix whose columns are s times the corresponding entries in A. In other words, you take the scalar and multiply it with every number in the matrix. Theorem: If r and s are scalars, then (r + s)A = rA + sA and r(sA) = (rs)A.
5. 5. Matrix Multiplication Before you multiply a matrix by another matrix, you need to check the compatibility of the two matrices. The number of columns (n) in A must equal the number of rows (m) in B in order to carry out the matrix multiplication. Example 1 : A = 2 5 B = 3 1 7 1 3 8 2 4 m * n m * n 2 * 2 2 * 3 You can see that the number of columns of A matches the number of rows of B. (Later on, you will see that the size of matrix AB will be m of A n of B). Example 2 : A = 2 5 B = 3 1 7 1 3 8 2 4 4 1 2 2 * 2 3 * 3 Obviously, n of A m of B and thus they are not compatible for multiplication.
6. 6. Definition: A collection of number (real or complex) are arranged in the from of a rectangular or square array (arrangement) in horizontal and vertical lines is called a Matrix. Row: The horizontal lines of number in a matrix are called row Columns: The vertical lines of number in a matrix are called columns 7 3 2 -5 8 2 -1 4 2 1-I 1+2I
7. 7. Order of matrix: The order of a matrix is defined in term of its number of rows and columns. Order of matrix = No of rows * No . of Columns 1 2 3 0 9 2
8. 8. Compact from of a matrix: A= where (< i< m) (< J < m)aji
9. 9. Determinants(Ad joint& Inverse of a matrix) Singular: A singular matrix is said to be a singular matrix if det A=0 EX: A = A =10 * 4-5 * 8 = 40-40=0 Non singular: A square matrix A is said to be a non singular if det A 0 EX: A = A =1 * 1-2 * 3 =1- 6 = -5 0 10 5 8 4 1 2 3 1
10. 10. Adjoins matrix Definition : If the element of a square matrix are replaced by elements then the transpose of the result matrix is called Ad jont matrix. A = A11 A12 A13 A21A22 A23 A31 A23 A33
11. 11. Inverse of matrix Definition: Let A be a non singular matrix. If there exists a square matrix B = AB =1 A 1 =AdjA/detA
12. 12. Invers matrix method Let A = X = B= X =A 1 B A 1 =1 /det * Adj A a1 b1 c1 a2 b2 c2 a3 b3 c3 X Y z d1 d2 d3
13. 13. Gauss-Jordan`s method Write down the Augmented Matrix: Apply Elementary Row transformations and reduce the Augmented matrix into the final matrix Then is the required solution. a1 b1 c1 d1 a2 b2 c2 d2 a3 b3 c3 d3 1 0 0 a 0 1 0 b 0 0 1 c x=a y=b z=c
14. 14. Steps of the Augmented matrix into Final Matrix & get solution: Step1: In the Final matrix the 1st column is So, first concentrate on 1st row and 1st column element in the Augmented matrix. 1 using k R1 (or)Ri Rj operation . Step2: 1st column of Final matrix. To get this 0 use R2 kR1 To get this 0 use R3 kR1 Step3: Now concentrate on 2nd column of Final matrix . Make the element in 2nd row &2nd column as 1 st using kR2 (or) R2R3 operation. 1 0 0 1 0 0
15. 15. Step4: 2nd column in final Matrix To get this 0 use R1- kR2 To get this 0 use R3- kR2 Step5: Now concentrate on 3rd of Matrix . make the element in 3rd Row &3rd column as 1 using kR3 operation. Step6: 3rd column of Final Matrix To get this 0 use R1 KR3 Step7: Get the solution from the element obtained in the last column of final Matrix x=a y=b z=c 0 1 0 0 0 1
16. 16. Cramer's rule Solving the given equation: a1x+b1y+c1z=d1 a2x+b2y+c2z=d2 a3x+b3y+c3z=d3 = a1 b1 c1 1 =d1 b1 c1 2 = a1 d1 c1 a2 b2 c2 0 d2 b2 c2 a2 d2 c3 a3 b3 c3 d3 b3 c3 a3 d3 c3 3 =a1 b1 d1 a2 b2 d2 a3 b3 d3 x=1/: y=2/: z=3/
17. 17. Royal CIVIL Presented by B.Lokeshwar.