Presentation on theme: "1.5 Elementary Matrices and a Method for Finding"— Presentation transcript:
11.5 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
2Examples: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.
3TheoremEvery 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
4To 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 | ]
6If 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.
71.6 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.
8Example: Solve the system by using LetWe showed in previous section thatThenThus x1 = 1, x2 = -1, x3 =2
9Linear 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)
10Example: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
11Theorem 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
12Theorem 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
13A 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.
14Determine 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:
15R2-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
16Example: 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