Matrix Definition An array of numbers in m

Matrix Definition: An array of numbers in m rows and n colums is called an mxn matrix A square matrix of order n, is an (nxn) matrix

Matrix Equality Two matrices A = [aij] and B = [bij] are equal if they have the same order and aij = bij for every i and j. For example, . since both matrices are of order 2 x 2 and all corresponding entries are equal.

Matrix Addition Definition: The sum of two matrices is the matrix where each element is the sum of the corresponding elements of the two matrices. NOTE: Matrix addition is only defined when the matrices have the same dimension. Examples:

Matrix Substraction Definition: The substraction of two matrices is the matrix where each element is the difference of the corresponding elements of the two matrices. NOTE: Matrix substraction is only defined when the matrices have the same dimension. Example:

Matrix Addition Theorem 1: Let A, B, and C be matrices of the same size, then Commutative law of addition Associative law of addition

Scalar Multiplication • Definition: Let a be a number and A=(aij)mxn be any matrix. Then • This means that a. A is the matrix by multiplying each entry of A by the same number a

Negative of a Matrix • Definition: Let A=(aij) be a mxn matrix. The negative of the matrix A is the mxn matrix B=(bij) such that bij=-aij for all i, j. The negative of A is written as –A. For example,

Zero Matrix • Definition: The mxn matrix in which every entry is 0 is called the mxn zero matrix and is denoted by 0 mxn or simply 0. • We note that A+(-A)=0, the zero matrix. • We usualy write A+(-B) as A-B

Multiplication of Matrices • The product of two matrices A and B is defined if A is mxn and B is nxp • We first give the rule for multplication in special case when A is 1 xn and B is nx 1 Then their product AB is defined to be a 1 x 1 matrix,

Multiplication of Matrices Now we give the rule for multiplication of matrices in general. Definition: Let A=(aij)mxn and B=(bij)nxq. Then AB =(cij)mxp

Multiplication of Matrices 1 2 3 4 5 6 10 20 30 12 22 32 1*10+2*20+3*30 1*12+2*22+3*32 4*10+5*20+6*30 4*12+4*22+6*32 # of Columns of A must = # of Rows of B

Diagonal of a Matrix Definition: If A is an nxn matrix, then the line joining (1, 1) entry, (2, 2) entry, …, (n, n) entry is called the diagonal (or the main diagonal) of the matrix. Definition: Any nxn matrix D is called diagonal matrix if each entry not on the diagonal is 0. For example,

Identity Matrix Definition: An nxn matrix A=(aij), where aij=1 whenever i=j, and aij=0 whenever is called the identity matrix of order n. And it is denoted by Square matrix with ones on the diagonal and zeros elsewhere. It can be easily proved that if A is an mxn matrx, then

Properties of Matrix Multiplication

Transpose Definition: Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by AT, whose columns are formed from the corresponding rows of A. In other words, the ith row of AT is the ith column of A for all i. Example: Let What is AT ?

Rules related to transpose:

Matrix Powers • Recall that matrix multiplication is associative, i. e. if A, B and C have the proper dimensions, then A ( BC ) = ( AB ) C, so the parentheses are unnecessary and the product can be written as ABC. • If A is an n x n matrix and p is a positive integer, can define • Again, if A is an n x n matrix, adopt the convention

Triangular Matrices • An n x n matrix A = [ aij ] is called upper triangular if aij = 0 for i > j • An n x n matrix A = [ aij ] is called lower triangular if aij = 0 for i < j • Note: – A diagonal matrix is both upper and lower triangular – The n x n zero matrix is both upper and lower triangular

Symmetry • Defn - A matrix A is called symmetric if AT= A • Defn - A matrix A is called skew-symmetric if AT= -A • Comment - If A is skew-symmetric, then the diagonal elements of A are zero • Comment - Any square matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix