Presentation on theme: "1.5 Elementary Matrices and a Method for Finding"— Presentation transcript:
1 1.5 Elementary Matrices and a Method for Finding An elementary row operation (sometimes called just a row operation)on a matrix A is any one of the following three types of operations:Interchange of two rows of AReplacement of a row r of A by c r for some number c ≠ 0Replacement of a row r1 of A by the sum r1 + c r2 of that row and amultiple of another row r2 of AAn n×n elementary matrix is a matrix produced by applying exactlyone elementary row operation to InEij is the elementary matrix obtained by interchanging the i-th and j-th rows of InEi(c) is the elementary matrix obtained by multiplying the i-th row of In by c ≠ 0Eij(c) is the elementary matrix obtained by adding c times the j-th row to the i-th row of In, where i ≠ j
2 Examples:Theorem (Row Operations by Matrix Multiplication)Suppose that E is an m×m elementary matrix produced by applying a particular elementary row operation to Im, and that A is an m×n matrix. Then EA is the matrix that results from applying that same elementary row operation to ARemark:The above theorem is primarily of theoretical interest. Computationally, it is preferable to perform row operations directly rather than multiplying on the left by an elementary matrix.
3 TheoremEvery elementary matrix is invertible, and the inverse is also an elementary matrix.Theorem (Equivalent Statements)If A is an n×n matrix, then the following statements are equivalent, that is, all true or all falseA is invertibleAx = 0 has only the trivial solutionThe reduced row-echelon form of A is InA is expressible as a product of elementary matrices
4 To find the inverse of an invertible matrix A, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtainExample: Using Row Operations to Find the inverse ofSolution:To accomplish this we shall adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A | I ]We shall apply row operations to this matrix until the left side is reduced to I; these operations will convert the right side to , so that the final matrix will have the form [I | ]
6 If and n X n matrix A is not invertible, then it cannot be reduced to In by elementary row operations, i.e, the computation can be stopped.Example:Row operationsrrefSince we have obtained a row of zeros on the left side, A is not invertible.
7 1.6 Further Results on Systems of Equations and Invertibility Theorem 1.6.1Every system of linear equations has either no solutions, exactly one solution, or in finitely many solutions.Theorem 1.6.2If A is an invertible n×n matrix, then for each n×1 matrix b, the system of equations Ax = b has exactly one solution, namely, x = b.Remark: this method is less efficient, computationally, than Gaussian elimination,But it is important in the analysis of equations involving matrices.
8 Example: Solve the system by using LetWe showed in previous section thatThenThus x1 = 1, x2 = -1, x3 =2
9 Linear Systems with a Common Coefficient Matrix To solve a sequence of linear systems, Ax = b1, Ax = b2, …, Ax = bk, with common coefficient matrix AIf A is invertible, then the solutions x1 = b1, x2 = b2 , …, xk = bkA more efficient method is to form the matrix [ A | b1 | b2| … | bk ], thenreduce it to reduced row-echelon form we can solve all k systems atonce by Gauss-Jordan elimination (Here A may not be invertible)
10 Example:Solve the systemSolution:Reducing this to reduced row-echelon form yields (details on board)Thus, the solution of system (a) is x1=1, x2=0, x3=1 and the solutionOf the system (b) is x1=2, x2=1, x3= -1
11 Theorem 1.6.3Let A be a square matrix(a) If B is a square matrix satisfying BA = I, then B =(b) If B is a square matrix satisfying AB = I, then B =Theorem 1.6.5Let A and B be square matrices of the same size. If AB is invertible, then A and B must also be invertible
12 Theorem 1.6.4 (Equivalent Statements) If A is an n×n matrix, then the following statements are equivalentA is invertibleAx = 0 has only the trivial solutionThe reduced row-echelon form of A is InA is expressible as a product of elementary matricesAx = b is consistent for every n×1 matrix bAx = b has exactly one solution for every n×1 matrix b
13 A Fundamental Problem: Let A be a fixed mXn matrix A Fundamental Problem: Let A be a fixed mXn matrix. Find all mX1 matrices b suchSuch that the system of equations Ax=b is consistent.If A is an invertible matrix, then for every mXn matrix b, the linear system Ax=b hasThe unique solution x= b.If A is not square, or if A is a square but not invertible, then theorem does notApply. In these cases the matrix b must satisfy certain conditions in order for Ax=bTo be consistent.
14 Determine Consistency by Elimination Example: What conditions must b1, b2, and b3 satisfy in order for the system ofequationsTo be consistent?Solution: The augumented matrix isWhich can be reduced to row-echelon form as follows:
15 R2-R1R3-2R1-R2R3+R2From the third row in the matrix that the system has a solution if and only ifb1, b2, and b3 satisfy the conditionB3 - b1 - b2 = 0 or b3 = b1 + b2
16 Example: What conditions must b1, b2, and b3 satisfy in order for the system of equationsTo be consistent?Solution: The augumented matrix isWhich can be reduced to row-echelon form :In this case, there are no restrictions on b1, b2 and b3. That is, the systemHas the unique solutionx1=-40b1+16b2+9b3, x2=13b1-5b2-3b3, x3=5b1-2b2-b3