【正文】 alent to A if B can be obtained from A by a finite number of row operations. In particular, two augmented matrices (A|b) and (B|c) are equivalent if and if AX=b and Bx=c are equivalent systems.Two properties of row equivalent matrices: (of course A is equivalent to itself, reflexivity 自反性)I. If A is equivalent to B, the B is equivalent to A (symmetry對稱性)II. If A is equivalent to B, and B is equivalent to C, then A is equivalent to C. (Transitivity 傳遞性)Theorem (Equivalent Conditions for Nonsingularity) Let A be an nxn matrix. The following are equivalent:(a) A is nonsingular. (b) AX=0 has only the trivial solution 0.(c) A is row equivalent to I.Proof (a) implies (b) Multiply both sides of the equation by the inverse of A.(two systems are equivalent)(b) implies (c)Use row operations to transform the system into the form UX=0, where U is in row echelon form. If one of the diagonal elements of U were 0, the last row of U would consist entirely of 0’s. But then AX=0 would be equivalent to a system with at least free variable and hence would have a nontrivial solution. Thus U must be a upper triangular matrix with diagonal elements all equal to 1. It follows then that I is the reduced row echelon form of A and hence A is now equivalent to I.(d) implies (a) If A is row equivalent to the identity matrix I, then A can be written as a product of a finite sequence of elementary matrices. All elementary matrices are invertible, so the product is also invertible. Hence, A is nonsingular.Corollary An nxn system AX=B has a unique solution if and only if A is nonsingular. Proof If A is nonsingular, then by premultiplying both sides of the equation by the inverse of A, and conclude that the solution must be equal to . Conversely, if AX=B has a unique solution X, then we claim A cannot be singular. Indeed, if A were singular, then the equation AX=0 would have a nontrivial solution Z (theorem ). But this imply that Y=X+Z is a second solution to AX=B. Therefore, A must be nonsingular. Theorem actually tells us a way to find the inverse of A if A is nonsingular. A is nonsingular if and only if A is row equivalent to the identity matrix I, and hence there are elementary matrices such that Thus .This implies that the same series of elementary row operations that transforms a nonsingular matrix into I will transform I into . Thus, if we augment A by I and perform the elementary row operations that transform A into I on the augmented matrix, then I will be transformed into .Example Compute if 224。3. Matrix AlgebraNew words and phrases:Algebra 代數(shù)Scalar 數(shù)量，標量Scalar multiplication 數(shù)乘Real number 實數(shù)Complex number 復數(shù)Vector 向量Row vector 行向量Column vector 列向量Euclidean nspace n維歐氏空間Linear bination 線性組合Zero matrix 零矩陣Identity matrix 單位矩陣Diagonal matrix 對角矩陣Triangular matrix 三角矩陣Upper triangular matrix 上三角矩陣Lower triangular matrix 下三角矩陣Transpose of a matrix 矩陣的轉置(Multiplicative ) Inverse of a matrix 矩陣的逆Singular matrix 奇異矩陣Singularity 奇異性Nonsingular matrix 非奇異矩陣Nonsingularity 非奇異性The term scalar (標量，數(shù)量) is referred to as a real number (實數(shù)) or a plex number（復數(shù)）. Matrix notationAn mxn matrix, a rectangular array of mn numbers. VectorsMatrices that have only one row or one column are of special interest since they are used to represent solutions to linear systems. We will refer to an ordered ntuple of real numbers as a vector（向量）. If an ntuple is represented in terms of a 1xn matrix, then we will refer to it as a row vector. Alternatively, if the ntuple is represented by an nx1 matrix, then we will refer to it as a column vector.In this course, we represent a vector as a column vector. The set of all nx1 matrices of real number is called Euclidean nspace (n 維歐氏空間) and is usually denoted by . Given a mxn matrix A, it is often necessary to refer to a particular row or column. The matrix A can be represented in terms of either its column vectors or its row vectors. or Equality For two matrices to be equal, they must have the same dimensions and their corresponding entries must agree★Definition: Two mxn matrices A and B are said to be equal if for each ordered pair (i, j) Scalar Multiplication If A is a matrix, is a scalar, then A is the mxn matrix formed by multiplying each of the entries of A by .★Definition: If A is an mxn matrix, is a scalar, then A is the mxn matrix whose (i, j) is for each ordered pair (i, j) . Matrix Addition Two matrices with the same dimensions can be added by adding their corresponding entries. ★Definition: If A and B are both mxn matrices, then the sum A+B is the mxn matrix whose (i,j) entry is for each ordered pair (i, j). An mxn zero matrix (零矩陣) is a matrix whose entries are all zero. It acts as an additive identity on the set of all mxn matrices.A+O=O+A=AThe additive of A is (1)A since A+(1)A=O=(1)A+A.AB=A+(1)BA=(1)A Matrix Multiplication and Linear Systems MotivationsRepresent a linear system as a matrix equationWe have yet to defined the most important operation, the multiplications of two matrices. A 1x1 system can be written A scalar can be treated as a 1x1 matrix. Our goal is to generalize the equation above so that we can represent an mxn system by a single equation.Case 1: 1xn systems If we set and , and define Then the equation can be written as 。(ii) Perform elementary ro