DEFINITION 1 Matrices 矩阵 A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n matrix The plural of matrix is matrices A matrix with the same number of rows as columns is called equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal
M a t r i c e s 矩 阵 DEFINITION 1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m × n matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal
DEFINITION 2 Matrices 矩阵 The ith row ofa is the 1 x n R 12 Let A= matrix ail, aj,., ain. The a. a.a. jth column of a is then X1 matrix The (i, j)th element or entry of A is the element aii that is, the number in the ith row and jth column ofA. A convenient shorthand notation for expressing the matrix A is to write A=ail, which indicates that a is the matrix with its (i, j)th element equal to a i
M a t r i c e s 矩 阵 DEFINITION 2. Let The (i, j)th element or entry of A is the element aij, that is, the number in the ith row and jth column of A. A convenient shorthand notation for expressing the matrix A is to write A = [aij], which indicates that A is the matrix with its (i, j)th element equal to aij. The ith row of A is the 1 × n matrix [ai1, ai2, …, ain]. The jth column of A is the n × 1 matrix
DEFINTION3 Matrices 矩阵 Leta=ai and b=bi be m X n matrices. The sum of A and b, denoted by a+ B, is the m x n matrix that has ai t bi as its (i,j)th element. In other words,A+ b ai +bil
M a t r i c e s 矩 阵 DEFINITION 3. Let A = [aij] and B = [bij] be m × n matrices. The sum of A and B, denoted by A + B, is the m × n matrix that has aij + bij as its (i, j)th element. In other words, A + B = [aij +bij]
DEFINITION4 Matrices 矩阵 Let a be an m x k matrix and b be a k x n matrix The product of A and B, denoted by ab, is the m X n matrix with (i, jth entry equal to the sum of the products of the corresponding elements from the ith row of a and the jth column of B In ab=cil, then 可=a1b1+ab2+…,+akbk
M a t r i c e s 矩 阵 DEFINITION 4. Let A be an m × k matrix and B be a k × n matrix. The product of A and B, denoted by AB, is the m × n matrix with (i, j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In AB = [cij], then Cij = ai1b1j + ai2b2j + … + aikbkj =
DEFINITION 5 Matrices 矩阵 The identity matrix of order n is the n X n matrix I i &,I, where al if i=j and 8-0 if j. Hence 01 了 00
M a t r i c e s 矩 阵 DEFINITION 5. The identity matrix of order n is the n × n matrix In = [ ], where =1 if i = j and = 0 if i ≠ j. Hence