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## MATRIX

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**CONCEPT OF MATRIX**• 1.Definition of matrix Equations Rectangular number table can be formed from the coefficients**This is a matrix**The rectangular number table with m rows and n columns constructed by mn numbers in a certain order is called a mn Matrix (briefly matrix). The horizontals are called the rows of matrix and the verticals are called the columns of matrix Is called the element located row i and column j. The matrix is called real matrix if its elements are real numbers.From now on, we only discuss the real matrix.**Matrix is usually denoted with capital A、B、**C and so on. For instance Simply write: Column matrix Footmark Row matrix**The main**diagonal line The matrix is called a square matrix if its row number is equal to its column number, i.e., m=n.**6.Trapezoid matrix Let**If holds for i>j (i<j), and the number of zero before (behind) the first (last) nonzero element is growing larger (less) as the rows increase, then we call A a upper (lower) trapezoid matrix. They are all called trapezoid matrixes.**???**No! Are they Trapezoid matrixes? Please remember the characteristics of trapezoid matrix and follow its definition. Trapezoid matrix is the most frequently-used matrix.**Operation of matrix**一、linear operation 1.Equality: equality of two matrixes means that their number of rows and number of columns are respectively equal and their corresponding elements are the same. i.e., = corresponding elements are the same same type Type is the same**2.addition、subtraction**define Let and Obviously, A+B=B+A (A+B)+C=A+(B+C) A+O=O+A=A A-A=O Negative matrix: the negative matrix of is We write-A, i.e.,**3.number multiple**Is called the product of number and matrix， Briefly number multiple。write：kA**=**In general，we have**what condition can enable A and B to multiply?**= O Obviously It is the difference between Matrix and number.**Example2**But Please remember： 1.The multiplication of matrixes does not satisfy commutative law; 2. does not satisfy cancellation law； 3. Has nonzero zero factor. This is another difference between matrix and number.**Please particularly**notice property 5. The conclusion does not hold if the orders of A and B are not equal.**Positive integer power of the square matrix.**Transposition of matrix. Please remember!**=**= That is**Symmetric and anti-symmetric matrix**Any square matrix can be decomposed to the sum of a symmetric matrix and an anti-symmetric matrix. The determinant of an odd order anti-symmetric matrix equals to zero. 0