2.  Row and Reduced Row Echelon  Elementary Matrices

4.  If m and n are positive integers, then an m  n matrix is a rectangular array in which each entry a ij of the matrix is a number. The matrix has m rows and n columns.

5.  A real matrix is a matrix all of whose entries are real numbers.  i (j) is called the row (column) subscript.  An m  n matrix is said to be of size (or dimension ) m  n.  If m=n the matrix is square of order n.  The a i,i ’s are the diagonal entries.

6.  Given a system of equations we can talk about its coefficient matrix and its augmented matrix.  These are really just shorthand ways of expressing the information in the system.  To solve the system we can now use row operations instead of equation operations to put the augmented matrix in row echelon form.

7. 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row.

8.  Two matrices are said to be row equivalent if one can be obtained from the other using elementary row operations.  A matrix is in row-echelon form if: › All rows consisting entirely of zeros are at the bottom. › In each row that is not all zeros the first entry is a 1. › In two successive nonzero rows, the leading 1 in the higher row is further left than the leading 1 in the lower row.

9. 1. Write the augmented matrix of the system. 2. Use elementary row operations to find a row equivalent matrix in row-echelon form. 3. Write the system of equations corresponding to the matrix in row- echelon form. 4. Use back-substitution to find the solutions to this system.

10.  In Gauss-Jordan elimination, we continue the reduction of the augmented matrix until we get a row equivalent matrix in reduced row- echelon form. (r-e form where every column with a leading 1 has rest zeros)

11.  A system of linear equations in which all of the constant terms is zero is called homogeneous.  All homogeneous systems have the solutions where all variables are set to zero. This is called the trivial solution.

12. Using Elementary Matrices

13.  An n by n matrix is called an elementary matrix if it can be obtained from I n by a single elementary row operation.  These matrices allow us to do row operations with matrix multiplication.

14. Theorem: Let E be the elementary matrix obtained by performing an elementary row operation on I n. If that same row operation is performed on an m by n matrix A, then the resulting matrix is given by the product EA.

15.  These correspond to the three types of EROs that we can do: › Interchanging rows of I -> Type I EM › Multiplying a row of I by a constant -> Type II EM › Adding a multiple of one row to another -> Type III EM

16.  E1 =  How is this created? Eg. 1 Suppose A =  E1A =  =  What is AE1?

17.  E2 =  E2A = =  AE2 = =

18.  E3 =  E3A = =  AE3 = =

19.  Let A and B be m by n matrices. Matrix B is row equivalent to A if there exists a finite number of elementary matrices E 1, E 2,... E k such that B = E k E k-1... E 2 E 1 A.

20.  This means that B is row equivalent to A if B can be obtained from A through a series of finite row operations.  If we then take two augmented matrices (A|b) and (B|c) and they are row equivalent, then Ax = b and Bx=c must be equivalent series

21.  If A is row equivalent to B, B is row equivalent to A  If A is row equivalent to B and B is row equivalent to C then A is row equivalent to C

22.  Compute the inverse of A for A =

23. Now, solve the system:

24.  We can employ the format Ax = b so x=A -1 b  We just calculated A -1 and b is the column vector  So we can easily find the values of x by multiplying the two matrices

25. Keys to calculating Inverses

26.  Require square matrices  Each square matrix has a determinant written as det(A) or |A|  Determinants will be used to: › characterize on-singular matrices › express solutions to non-singular systems › calculate dimension of subspaces

27.  If A and B are square then  It is not difficult to appreciate that  If A has a row (or column) of zeros then  If A has two identical rows (or columns) then

28.  If B is obtained from A by ERO, interchanging two rows (or columns) then  If B is obtained from A by ERO where row (or column) of A were multiplied by a scalar k, then

29.  If B is obtained from A by ERO where a multiple of a row (or column) of A were added to another row (or column) of A then

30. That Is the determinant is equal to the product of the elements along the diagonal minus the product of the elements along the off-diagonal.

31. Note: The matrix A is said to be invertible or non-singular if det(A)≠ 0. If det(A) = 0, then A is singular.

32.  Using EROs on rows 2 and 3

33.  The matrix will be row equivalent to I iff:

34.  This implies that the Det(A) =

35.  Use EROs to find:

36.  STEP 1: Apply from property 5 this gives us  STEP 2: Convert matrix to Echelon form

37.  Therefore is the same as: matrix is now in echelon form so we can multiply elements of main diagonal to get determinant

38.  Factorize the determinants of  What is ?

39.  We see that y – x is a factor of row 2 and z – x is a factor of row 3 so we factor them out from:  And we get:

41.  The matrix is now in echelon form so we can multiply elements of main diagonal to get determinant and then multiply by factors to get: =

42.  Now, the matrix corresponds to  Since =

43.  Then =

44.  Cofactor expansion is one method used to find the determinant of matrices of order higher than 2.

45.  If A is a square matrix, then the minor M i,j of the element a i,j of A is the determinant of the matrix obtained by deleting the ith row and the jth column from A.

46.  Consider the matrix. The minor of the entry “0” is found by deleting the row and the column associated with the entry “0”.

47.  The minor of the entry “0” is Note: Since the 3 x 3 matrix A has 9 elements there would be 9 minors associated with the matrix.

48.  The cofactor C i,j = (-1) i+j M i,j. Since we can think of the cofactor of as nothing more than its signed minor.

49.  Find the minor and cofactor of the entry “2” for  We first need to delete the row and column corresponding to the entry “2”

50.  The Minor of 2 is  The minor corresponds to row 1 and column 2 so applying the formula, we have  So the cofactor of the entry “2” is 40.

51.  Theorem: Let A be a square matrix of order n. Then for any i,j,  Columns: and  Rows:

52.  Given find det(A).  Cofactor is found for the first entry in column 1 “-3”  Cofactor is found for the second entry in column 1 “-5”

53.  Cofactor is found for the third entry in column 1 “5”  The cofactors are then multiplied by the corresponding entry and summed.

55.  Using row 2 - expansion we fix row 2 and find the minors for each entry in row 2 then apply the sign corresponding to each entries position to find the cofactors. The cofactors are then multiplied by the corresponding entry and summed. 

57.  It is easy to show that

58.  If A is square and is in Echelon form then is the product of the entries on the (main) diagonal.

59.  Using CRAMER’S RULE we can apply this method to finding the solution to a system of linear systems that have the same number of variables as equations  There are two cases to consider

60.  Consider the square system AX = B where A is n x n.  If then the system has either I) No solution or ii) Many solutions  If A 1 is formed from A by replacing column 1 of A with column B and  I., then the system has NO solution  II. |A| ≠ 0, then the system has a unique solution

61.  Consider the square system AX = B where A is n x n.  If |A|≠ 0 then the system has a unique solution  The unique solution is obtained by using the Cramer’s rule.

62.  Where A i is found from A by replacing column i of A with B.

63.  Use Cramer’s rule to write down the solution to the system

65.  The adjoint of A (adj.A)) is- :, the transpose of the matrix of cofactors (a matrix of signed minors).  If then the inverse of A exists and  If, A has no inverse.

66.  Find inverse of  Det(A) = = (3*2*0) – (3*3*1) – (-2*1*0)+(-2*1*1) +(3*1*3) –(3*1*2) =0 – 9 – 0 – 2 + 9 – 6 = -8

69.  Given that show: a. b. c. d. e. Solve where and

71.  Taking the determinant  Implying that and therefore; exists

72.  Since,  Multiplying through by A

73.  Multiplying through by A -1

75.  Since