【正文】 same procedure is repeated for columns 3 through n1.Note that at the second step, row 1 and column 1 remain unchanged, at the third step, the first two rows and first two columns remain unchanged, and so on.At each step, the overall dimensions of the system are effectively reduced by 1. (The number of equations and the number of unknowns all decrease by 1.)If the elimination process can be carried out as described, we will arrive at an equivalent strictly triangular system after n1 steps.However, the procedure will break down if all possible choices for a pivot element are all zero. When this happens, the alternative is to reduce the system to certain special echelon form（梯形矩陣）.Assignment Students should be able to do all problems.Handin problems are: 711167。Put the free variables on the righthand side, it follows thatThus for any real numbers and , the 5tupleis a solution.Thus all ordered 5tuple of the form are solutions to the system. Reduced Row Echelon Form★Definition: A matrix is said to be in reduced row echelon form if :(i) the matrix is in row echelon form.(ii) The first nonzero entry in each row is the only nonzero entry in its column. The process of using elementary row operations to transform a matrix into reduced echelon form is called GaussianJordan reduction.The procedure for solving a linear system:(i) Write down the augmented matrix associated to the system。Case 2： mxn systems Consider an mxn system, and let A be the coefficient matrix, X the vector of unknowns, and B the vector of constants on the righthand side, and define the product AX by ….., then the linear system is equivalent to the matrix equation AX=B. Matrix Multiplication More generally, it is possible to multiply a matrix A times a matrix B if the number of columns of A equals the number of rows of B. ★Definition If is an mxn matrix and is an nxr matrix, then the product is the mxr matrix whose entries are defined by An alternative way to represent the linear system as a matrix equation is to express the product AX as a sum of vectors.★Definition are vectors in , and are scalars, then a sum of the form is said to a linear bination of the vectors . It follows that the product AX is a linear bination of the columns vector of A. provides a nice way of characterizing whether a linear system of equations is consistent. Theorem (Consistency Theorem for Linear Systems) A linear system AX=B is consistent if and only if B can be written as a linear bination of the column vectors of A. Example 6 on page 37 Notational RulesIf an expression involves both multiplication and addition and there are no parentheses to indicate the order of operations, multiplications are carried out before additions. This is true for both scalar and matrix multiplication. Algebraic Rules Theorem Each of the following statements is valid for any scalars and and for any matrices A, B, and C for which the indicated operations are defined.Properties 191. (Commutative law of addition)2. (Associative law of addition)3. (Associative law of matrix multiplication)4. (Left distributive law)5. (Right distributive law)6. 7. 8. 9. (Distributive law)We prove the associative law of matrix multiplication.The details of the proof will be given in class.Warning: In general, Matrix multiplication is not mutative.Notation: . Diagonal and Triangular MatricesAn nxn matrix A is said to be upper triangular（上三角的） if for ij and lower triangular （下三角的） if for ij. Also, A is said to triangular if it is either upper triangular of lower triangular. A 4x4 upper triangular matrix A 4x4 lower triangular matrix An nxn matrix is said to be diagonal（對(duì)角的） if whenever .A 3x3 diagonal matrix The Identity MatrixJust as the number 1 acts an identity for the multiplications of real numbers, there is a special matrix I that acts as an identity for matrix multiplication, that is IA=AI=A for any nxn matrix A.★Definition The nxn identity matrix（單位矩陣） if the matrix , whereA 3x3 identity matrix Matrix Inversion A real number a is said to have a multiplicative inverse (乘法逆) if there exists a number b such that ab=1. Any nonzero number a has a multiplicative inverse b=1/a. We generalize the concept of multiplicative inverse to matrices.★Definition: An nxn matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I. The matrix B is said to be a multiplicative inverse (or simply inverse) of A. The multiplicative inverse of a matrix A is unique. We denote it by .★Definition An nxn matrix A is said to be singular if it does not have a multiplicative inverse.Question: Does the matrix have a multiplicative inverse?Theorem If A and B are nonsingular nxn matrices, then AB is also nonsingular and .The proof will be given in class. The Transpose of a MatrixGiven an mxn matrix A, it is often useful to form a new nxn matrix whose columns are the rows of A.★Definition The transpose （轉(zhuǎn)置） of an nxm matrix A is the nxm matrix B defined by for j=1, 2, …, n and i=1, 2, …, m. The transpose of is denoted by Algebraic Rules for Transpose1. 2. 3. 4. We prove the 4th property in class.★Definition An nxn matrix A is said to be symmetric（對(duì)稱的） if .A 5x5 symmetric matrix: AssignmentNot required problems 30, 31Handin problems: 11. 12. 13. 15. 16. 17. 22. 24. 27. 167