【正文】 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。(ii) Perform elementary row operations to reduce the augmented matrix into a row echelon form。2. Row Echelon FormNew words and phrases:Row echelon form 行階梯形Reduced echelon form 簡化階梯形Lead variable 首變量Free variable 自由變量Gaussian elimination 高斯消元GaussianJordan reduction. 高斯若當(dāng)消元Overdetermined system 超定方程組Underdetermined system Homogeneous system 齊次方程組Trivial solution 平凡解 Examples and DefinitionIn this section, we discuss how to use elementary row operations to solve mxn systems.Use an example to illustrate the idea.★ Example: Example 1 on page 13. Consider a system represented by the augmented matrix224。(2) Multiply through one equation of a system by a nonzero real number。Chapter 1 Matrices and Systems of EquationsLinear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社會學(xué)）, ecology（生態(tài)學(xué)）, demography(人口統(tǒng)計(jì)學(xué)), and genetics（遺傳學(xué)）, etc.167。 (The solution set is empty)(3) two lines coincide. (The solution set has infinitely many elements)The situation is the same for mxn systems. An mxn system may not be consistent. If it is consistent, it must either have exactly one solution or infinitely many solutions. These are only possibilities.Of more immediate concerns is the problem of finding all solutions to a given system. Equivalent systems Two systems of equations involving the same variables are said to be equivalent （等價(jià)的，同解的）if they have the same solution set. To find the solution set of a system, we usually use operations to reduce the original system to a simpler equivalent system.It is clear that the following three operations do not change the solution set of a system.(1) Interchange the order in which two equations of a system are written。(3) Replace a row by its sum with a multiple of another row.Remark: The importance of these three operations is that they do not change the solution set of a linear system and may reduce a linear system to a simpler form. An example is given here to illustrate how to perform row operations on a matrix. ★ Example: The procedure for applying the three elementary row operations:Step 1: Choose a pivot element (主元)(nonzero) from among the entries in the first column. The row containing the pivot number is called a pivotal row（主行）. We interchange the rows (if necessary) so that the pivotal row is the new first row. Multiples of the pivotal row are then subtracted form each of the remaining n1 rows so as to obtain 0’s in the first entries of rows 2 through n. Step2: Choose a pivot element from the nonzero entries in column 2, rows 2 through n of the matrix. The row containing the pivot element is then interchanged with the second row ( if necessary) of the matrix and is used as the new pivotal row. Multiples of the pivotal row are then subtracted form each of the remaining n2 rows so as to eliminate all entries below the pivot element in the second column.Step 3: The