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Linear Algebra
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This video covers the basics of Linear Algebra (Matrices) and might be helpful in the Last minute revision for the GATE examination

Pawan Bhatla
Currently working for an Engineering Consultancy. I have completed my Instrumentation Engineering from Swami Vivekanand college. Educating i

Satish chandra
7 months ago
thanks
1. Linear Algebra

2. Content 1.Types of Matrices. 2.Matrix Algebra. 3.Transpose and Inverse of a Matrix. 4.Rank of a Matrix. 5 Consistency of Linear system of equations. 6.Eigen values and properties of Eigen values.

3. Types of matrices 1 2 31 5 6 17 8 9 1.Square Matrix A matrix with same number of rows and column 2.Diagonal matrix All the elements are zero except the diagonal elements A-0 2 0

4. Types of matrices 30 0 0 3.Scalar Matrix |A=10 a 0 0 Diagonal Matrix whoes diagonal elements are Equal o 0 3 0 0 0 0 3 1 0 0 O 0 1O O 0 01 O 0 00 1 4.Identity Matrix Diagonal elements are unity

5. 5.Null Matrix All elements are zero 6.Triangular Matrix a.Upper Triangular b. Lower Triangular 1 2 3 12B 05 6 0 4 4 5 6 0 0 9 7 8 9 10

6. Types of Matrices 0 1 0 1 8.Idempotent Matrix00 0 7.Peroidic Matrix Akt1= Ak A = 0 0 -9A3 = 03 A is nilpotent of order 3 A2-A (2-24) 514) 3 8 | B4-B B is periodic of order 3 A ion 9.Involutory Matrix B- 1 AkI 10.Nilpotent Matrix 0 01 Ak=0 C-C C is idempotent

7. Algebra of Matrice:s 1.Addition of Matrices. 1 2 3 4 5 6 7 8 1+5 2+6 3+74+ 8 6 10 12 8

8. Algebra of Matrices 2.Substraction of Matrices. 3 4 2 2 0 3 1 3 0 5 8 4 2 4 1 2 4 2 4 3 2

9. Multiplication of Matrices a Scalar Multiplication I-) A(BC) = (AB)(' --) A(B + C) = AB + AC. 3-) (B+C)A=BA + CA 4-) r(AB)=(rA)B=4(1B) for any scalar r 5-) Ind = A Aln for m n matrix A

10. Multiplication of Matrices b.Matrix Multiplication 1 -2 0 3 4 -5 2 x3 , 11 8 15 13 34-30 -12 -46 35 3 x 3 3 x 2

11. Transpose of a Matrix Interchange all the rows and column a11 a21 a31 a11 a12 a13 a12 a22 a32 a21 a22 a23 a13 a23 a33 a31 a32 a33

12. Transpose of a Matrix Example, Transpose of a matrix

13. Application of a Transpose 1.Symmetric and Skew-Symmetric Matrix .Symmetric: AT-A Skew-symmetric: AT-A. . Examples: 1 1-1] 1 2O 0 1-2 2 -3 0 skew-symmetric 1 0 3 1 0 symmetric

14. Inverse of a Matrix Method 2: Adjoint Method Step1: Find the Determinant of the Matrix A ( det(A) must be non-zero). Step2: Find the Minors and Co-factors of each element and construct Co-factor Matrix Step3: Find Transpose of the Cofactor Matrix which will give Adjoint Matrix. Step4: Inverse of the Matrix- A-1- ac det(A)

15. Rank of a Matrix Example 2 -1 2 -1 2-1 2-1 2r2 dded to n O-1 O 1 C-1r -r, added to r 1 O 0 1 O 0

16. Rank of a Matrix Example 2 2 3 4 5 3 4 5 6 4 5 6 7 9 10 11 12 A= Solution: Reduce the matrix to echelon form, 2 3 4 5 3 456 4 5 6 7 9 10 1 12 1 O O O O 1 2 3

17. Consistency of Linear System of equations Inconsistent" "Consistent" "Independent" "Dependent" Yi ntersect No SolutionOne Solution oo Solutions

18. Eigen Values 5 2 21 A- 3 6 3 /669 21 001 0 6 6 9 0 0 66 9- 1 31-A| 0 [522] 5-A 2 A-11-13 6 1 3 6-A 5- det(A-A1) =| 3 6-A |=-X+20 2-93 126=-(A-3)2(2-14) 3 9-

19. Properties of Eigen Values Trace sum of eigenvalues . Determinant-product of eigenvalues Po. , leads to leads to Ak- . A is invertible for non-zero eigenvalues only . Invertible - power property holds for -1 . A is hermitian - eigenvalues are real A is unitary-eigenvalues satisfy | | = 1