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MATRICES MATRIX OPERATIONS About Matrices A matrix is a rectangular arrangement of numbers in rows and columns. Rows run horizontally and columns run vertically. The dimensions of a matrix are stated “m x n” where ‘m’ is the number of rows and ‘n’ is the number of columns. Equal Matrices Two matrices are considered equal if they have the same number of rows and columns (the same dimensions) AND all their corresponding elements are exactly the same. Special Matrices Some matrices have special names because of what they look like. a) Row matrix: only has 1 row. b) Column matrix: only has 1 column. c) Square matrix: has the same number of rows and columns. d) Zero matrix: contains all zeros. Matrix Addition You can add or subtract matrices if they have the same dimensions (same number of rows and columns). To do this, you add (or subtract) the corresponding numbers (numbers in the same positions). Matrix Addition Example: 2 4 1 0 5 0 2 1 1 3 3 3 3 4 7 1 2 0 Scalar Multiplication To do this, multiply each entry in the matrix by the number outside (called the scalar). This is like distributing a number to a polynomial. Scalar Multiplication Example: 2 4 8 16 4 5 0 20 0 1 3 4 12 Matrix Multiplication Matrix Multiplication is NOT Commutative! Order matters! You can multiply matrices only if the number of columns in the first matrix equals the number of rows in the second matrix. 2 columns 3 2 5 6 1 3 9 7 2 0 4 5 2 rows Matrix Multiplication Take the numbers in the first row of matrix #1. Multiply each number by its corresponding number in the first column of matrix #2. Total these products. 3 2 5 6 1 3 9 7 2 0 4 5 2 1 3 3 11 The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1; ... Matrix Multiplication Notice the dimensions of the matrices and their product. 3 2 5 6 1 3 9 7 3x2 __ 8 15 11 2 0 13 34 30 4 5 12 46 35 2 x__ 3 3 x__ 3 __ Matrix Multiplication Another example: 2 1 9 0 5 2 10 5 3x2 2x1 8 45 60 3x1 Matrix Determinants A Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant. The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| or 3 1 . 2 4 Matrix Determinants To find the determinant of a 2 x 2 matrix, multiply diagonal #1 and subtract the product of diagonal #2. Diagonal 2 = -2 3 1 2 4 Diagonal 1 = 12 12 (2) 14 Matrix Determinants To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then do 6 diagonal products. 18 60 16 5 2 6 5 2 2 1 4 2 1 3 3 4 3 3 -20 -24 36 Matrix Determinants The determinant of the matrix is the sum of the downwards products minus the sum of the upwards products. 18 60 16 5 2 6 5 2 2 1 4 2 1 3 3 4 3 3 -20 -24 = (-8) - (94) = -102 36 Identity Matrices An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else. 1 0 0 0 1 0 0 0 1 1 0 0 1 When you multiply a matrix by the identity matrix, you get the original matrix. Inverse Matrices When you multiply a matrix and its inverse, you get the identity matrix. 3 1 2 1 1 0 5 2 5 3 0 1 Inverse Matrices Not all matrices have an inverse! To find the inverse of a 2 x 2 matrix, first find the determinant. a) If the determinant = 0, the inverse does not exist! The inverse of a 2 x 2 matrix is the reciprocal of the determinant times the matrix with the main diagonal swapped and the other terms multiplied by -1. Inverse Matrices 3 1 Example 1: A 5 2 det(A) 6 (5) 1 1 2 1 2 1 A 1 5 3 5 3 1 Inverse Matrices Example 2: 2 2 B 5 4 det(B) (8) (10) 2 2 2 1 4 B 5 2 5 2 2 1 1 1