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Fundamental concept in matrix

- 1. MATRIX Baiju P Assistant Professor Department of Economics Kristu Jayanti College (Bengaluru) 11/10/2023
- 2. Introduction • A matrix is a fundamental concept in mathematics and various fields of science and engineering. It is a rectangular array of numbers, symbols, or expressions organized in rows and columns. Matrices are widely used to represent and manipulate data, perform mathematical operations, and solve systems of linear equations. KJC 11/10/2023 2
- 3. Basics • Rows and Columns: A matrix consists of rows and columns. The number of rows is called the "row dimension," and the number of columns is called the "column dimension." A matrix with 'm' rows and 'n' columns is referred to as an "m x n" matrix. • Elements: Each entry or element in a matrix is denoted by aij, where 'i' represents the row number and 'j' represents the column number. For example, a21 refers to the element in the second row and first column. KJC 11/10/2023 3
- 4. Fundamental rules, properties and types of matrices • Matrices obey various rules and properties that are important to understand when working with them in linear algebra and various mathematical applications. • Here are some fundamental rules and properties of matrices: 11/10/2023 KJC 4
- 5. Matrix Addition and Subtraction: 11/10/2023 KJC 5 If A and B are two matrices of the same size (having the same number of rows and columns), then they can be added or subtracted element-wise. The result is a new matrix C, where each element c_ij is the sum (or difference) of the corresponding elements from A and B.
- 6. Problems • Problem 1: 11/10/2023 KJC 6
- 7. Scalar Multiplication: You can multiply a matrix by a scalar (a single number) by multiplying every element of the matrix by that scalar. If A is a matrix and k is a scalar, then kA represents scalar multiplication. 11/10/2023 KJC 7
- 8. Matrix Multiplication: If A is an m×n matrix and B is an n×p matrix, then their product AB is defined, and the resulting matrix will be an m×p matrix. In matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix 11/10/2023 KJC 8
- 9. Associative Property of Matrix Multiplication: (AB)C = A(BC), where A, B, and C are matrices of appropriate sizes. KJC 11/10/2023 9
- 10. Distributive Property of Matrix Multiplication: A(B + C) = AB + AC, and (A + B)C = AC + BC, where A, B, and C are matrices of appropriate sizes. KJC 11/10/2023 10
- 11. Distributive Property of Matrix Multiplication: A(B + C) = AB + AC, and (A + B)C = AC + BC, where A, B, and C are matrices of appropriate sizes. KJC 11/10/2023 11