Arithmetic Operations on Matrices

1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row Matrix to Column Matrix 5. Matrix Multiplication 6. Identity Matrix 7. Matrix Equation

 A matrix is any rectangular array of numbers and may be of any size.  The size of a matrix is n x k where n is the number of rows and k is the number of columns.  The entry a ij refers to the number in the i th row and j th column of the matrix.  Two matrices are equal provided that they have the same size and that all their corresponding entries are equal. 3

 is a 2x3 matrix.  The entry a 1,2 = -1.  The entry a 2,3 = 7. 4

 A row matrix or row vector only has one row.  A column matrix or column vector only has one column.  A square matrix has the same number of rows as columns. 5

 is a 2x2 matrix and a square matrix. 6 is a 1x4 matrix and a row matrix. is a 3x1 matrix and a column matrix.

 The sum A + B of two matrices A and B is defined only if A and B are two matrices of the same size. In this case A + B is the matrix formed by adding the corresponding entries of A and B.  Two matrices of the same size are subtracted by subtracting corresponding entries. 7

8 is not defined.

 If A is a row matrix and B is a column matrix, then we can form the product A  B provided that the two matrices have the same length. The product A  B is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum. 9

10 is not defined.

 If A is an m x n matrix and B is an n x q matrix, then we can form the product A  B. The product A  B is an m x q matrix whose entries are obtained by multiplying the rows of A by the columns of B. The entry in the i th row and j th column of the product A  B is formed by multiplying the i th row of A and j th column of B. 11

is not defined.

 The identity matrix I n of size n is the n x n square matrix with all zeros except for ones down the upper-left-to-lower-right diagonal.  Here are the identity matrix of sizes 2 and 3: 13

14 For all n x n matrices A, I n  A = A  I n = A.

 The matrix form of a system of linear equations is  AX = B  where A is the coefficient matrix whose rows correspond to the coefficients of the variables in the equations. X is the column matrix corresponding to the variables in the system. B is the column matrix corresponding to the constants on the right-hand side of the equations. 15

16 Write the following system as a matrix equation Equation 1 Equation 2 x y constants

 A matrix of size m x n has m rows and n columns.  Matrices of the same size can be added (or subtracted) by adding (or subtracting) corresponding elements. 17

 The product of an m x n and an n x r matrix is the m x r matrix whose ij th element is obtained by multiplying the i th row of the first matrix by the j th column of the second matrix. (The product of each row and column is calculated as the sum of the products of successive entries.) 18