Matrix and It's Applications
- 1. Presentation on Matrix and it`sApplications Presented by Pritom Chaki Roll: 19021060 MBA (Professional) Batch-21 Bangladesh University of Professionals 1
- 2. Contents Definition of a Matrix Operations of Matrices Determinant of Matrix Inverse of Matrix Linear System to Matrix Unique Properties of Matrix Applications of Matrices 2
- 3. Matrix (Basic Definition) Matrices are the rectangular agreement of numbers, expressions, symbols which are arranged in columns and rows. A = 𝑎11 ⋯ 𝑎1𝑛 𝑎21 ⋯ 𝑎2𝑛 … 𝑎 𝑚1 … … … 𝑎 𝑚𝑛 = {𝐴𝑖𝑗} 3
- 4. Operations with Matrices (Sum, Difference) If A and B have the same dimensions, then their sum, A + B, is obtained by adding corresponding entries. In symbols, (𝑨 + 𝑩)𝒊𝒋 = 𝒂𝒊𝒋 + 𝒃𝒊𝒋. If A and B have the same dimensions, then their difference, A - B, is obtained by subtracting corresponding entries. In symbols, (𝑨 − 𝑩)𝒊𝒋 = 𝒂𝒊𝒋 - 𝒃𝒊𝒋. 4
- 5. Operations with Matrices (Sum, Difference) Sum: 3 4 1 6 7 0 + −1 0 7 6 5 1 = 2 4 8 12 12 1 The matrix 0 whose entries are all zero. Then, for all A , A + 0 =A Difference: 2 4 8 12 12 1 - 3 4 1 6 7 0 = −1 0 7 6 5 1 The matrix 0 whose entries are all zero. Then, for all A , A - 0 =A 5
- 6. Operations with Matrices (Scalar Multiple) If A is a matrix and r is a number (sometimes called a scalar in this context), then the scalar multiple, rA, is obtained by multiplying every entry in A by r. In symbols, (𝑟𝐴)𝑖𝑗 = 𝑟𝑎𝑖𝑗 . Example: 2 3 4 1 6 7 0 = 6 8 2 12 14 0 6
- 7. Operations with Matrices (Product) If A has dimensions k × m and B has dimensions m × n, then the product AB is defined, and has dimensions k × n. The entry (𝑨𝑩)𝒊𝒋 is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results i.e., Example: (𝑎𝑖1 𝑎𝑖2 … 𝑎𝑖𝑚) 𝑏1𝑗 𝑏2𝑗 … 𝑏 𝑚𝑗 = 𝑎𝑖1 𝑏1𝑗 + 𝑎𝑖2 𝑏2𝑗 + ⋯ +𝑎𝑖𝑚 𝑏 𝑚𝑗 7
- 8. Operations with Matrices (Product) Example: (𝒂𝒊𝟏 𝒂𝒊𝟐 … 𝒂𝒊𝒎) 𝒃 𝟏𝒋 𝒃 𝟐𝒋 … 𝒃 𝒎𝒋 = 𝒂𝒊𝟏 𝒃 𝟏𝒋 + 𝒂𝒊𝟐 𝒃 𝟐𝒋 + ⋯ +𝒂𝒊𝒎 𝒃 𝒎𝒋 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇 𝑨 𝑩 𝑪 𝑫 = 𝒂𝑨 + 𝒃𝑪 𝒂𝑩 + 𝒃𝑫 𝒄𝑨 + 𝒅𝑪 𝒄𝑩 + 𝒃𝑫 𝒆𝑨 + 𝒇𝑪 𝒆𝑩 + 𝒇𝑫 8
- 9. Operations with Matrices (Transpose) The transpose, 𝑨 𝑻 , of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an k×n matrix and B = 𝑨 𝑻 then B is the n×k matrix with 𝒃𝒊𝒋= 𝒂𝒋𝒊. If 𝑨 𝑻=A, then A is symmetric 𝒂 𝟏𝟏 𝒂 𝟏𝟐 𝒂 𝟏𝟑 𝒂 𝟐𝟏 𝒂 𝟐𝟐 𝒂 𝟐𝟑 𝑻 = 𝒂 𝟏𝟏 𝒂 𝟐𝟏 𝒂 𝟏𝟐 𝒂 𝟐𝟐 𝒂 𝟏𝟑 𝒂 𝟐𝟑 9
- 10. Laws of Matrix Algebra The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties. Associative Laws: A+ (B + C) = (A +B) + C, (AB)C = A(BC). Commutative Law for Addition: A + B = B + A Distributive Laws: A(B + C) = AB + AC, (A + B)C = AC + BC 10
- 11. Determinant of Matrix Determinant is a scalar Defined for a square matrix Is the sum of selected products of the elements of the matrix each product being multiplied by +1 or -1 𝑑𝑒𝑡 𝑎 𝑏 𝑐 𝑑 = ad - bc 11
- 12. Inverse of Matrix Definition: An inverse matrix 𝑨−𝟏 which can be found only for a square and a non-singular matrix A, is a unique matrix satisfying the relationship A 𝑨−𝟏 = 𝑰 = 𝑨−𝟏 A The formula for deriving the inverse is 𝑨−𝟏 = 𝟏 𝒅𝒆𝒕(𝑨) 𝒂𝒅𝒋(𝑨) 12
- 13. System of Equations in Matrix Form The system of linear equations: a11x1 + a12x2+a13x3+….+a1nxn = b1 a21x1 + a22x2+a23x3+….+a2nxn = b2 ……………………………………… ak1x1 + ak2x2+ak3x3+….+aknxn = bk Can be rewritten as the matrix equation Ax = b, where A = 𝑎11 … 𝑎1𝑛 𝑎21 … … … 𝑎2𝑛 … 𝑎 𝑘1 … 𝑎 𝑘𝑛 , x = 𝑥1 𝑥2 … 𝑥 𝑛 , b = 𝑏1 𝑏2 … 𝑏 𝑘 13
- 14. Example: Linear System to Matrix Equation = 4𝑥 + 𝑦 + 2𝑧 = 4 5𝑥 + 2𝑦 + 𝑧 = 4 𝑥 + 3𝑧 = 3 A = 4 1 2 5 2 1 1 0 3 , x = 𝑥 𝑦 𝑧 , b = 4 4 3 AX = d 4 1 2 5 2 1 1 0 3 𝑥 𝑦 𝑧 = 4 4 3 14
- 15. Unique Properties of Matrices In normal algebra , if we multiply two nonzero values, then the outcome will never be a zero . But if we multiply two non-zero values in matrix , then the outcome can be zero. Example: 𝑨 = 𝟑 𝟑 −𝟑 𝟑 and B= 𝟏 𝟏 𝟏 𝟏 AB = 𝟑 𝟑 −𝟑 𝟑 * 𝟏 𝟏 𝟏 𝟏 = 𝟑 ∗ 𝟏 + 𝟏 ∗ (−𝟑) 𝟑 ∗ 𝟏 + −𝟑 ∗ 𝟏 −𝟑 ∗ 𝟏 + 𝟑 ∗ 𝟏 −𝟑 ∗ 𝟏 + 𝟑 ∗ 𝟏 = 𝟑 − 𝟑 𝟑 − 𝟑 −𝟑 + 𝟑 −𝟑 + 𝟑 = 𝟎 𝟎 𝟎 𝟎 15
- 16. Application of Matrix Field of Geology Taking Seismic Surveys Plotting Graphs & Statistics Scientific Analysis Field of Statistics & Economics Presenting real world data such as People's habit, traits & survey data Calculating GDP 16
- 17. Application of Matrix Field of Animation Operating 3D software & Tools Performing 3D scaling/Transforming Giving reflection, rotation Physics & Electronics Elementary particles in quantum field theory Traditional mesh & nodal analysis Behavior of electronic components 17