Presentation on theme: "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."— Presentation transcript:
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 A Replacement of a row r of A by c r for some number c ≠ 0 Replacement of a row r1 of A by the sum r1 + c r2 of that row and a multiple of another row r2 of A An n×n elementary matrix is a matrix produced by applying exactly one elementary row operation to I n Eij is the elementary matrix obtained by interchanging the i-th and j- th rows of I n Ei(c) is the elementary matrix obtained by multiplying the i-th row of I n by c ≠ 0 Eij(c) is the elementary matrix obtained by adding c times the j-th row to the i-th row of In, where i ≠ j
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 A Remark: 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.
Theorem Every 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 false A is invertible Ax = 0 has only the trivial solution The reduced row-echelon form of A is I n A is expressible as a product of elementary matrices
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 I n to obtain Example: Using Row Operations to Find the inverse of Solution: 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 | ]
Row operations rref Thus
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 operations rref Since we have obtained a row of zeros on the left side, A is not invertible.
1.6 Further Results on Systems of Equations and Invertibility Theorem Every system of linear equations has either no solutions, exactly one solution, or in finitely many solutions. Theorem If 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.
Example: Solve the system by using Let We showed in previous section that Then Thus x 1 = 1, x 2 = -1, x 3 =2
To solve a sequence of linear systems, Ax = b 1, Ax = b 2, …, Ax = b k, with common coefficient matrix A Linear Systems with a Common Coefficient Matrix If A is invertible, then the solutions x 1 = b 1, x 2 = b 2, …, x k = b k A more efficient method is to form the matrix [ A | b 1 | b 2 | … | b k ], then reduce it to reduced row-echelon form we can solve all k systems at once by Gauss-Jordan elimination (Here A may not be invertible)
Example:Solve the system Solution: Reducing this to reduced row-echelon form yields (details on board) Thus, the solution of system (a) is x 1 =1, x 2 =0, x 3 =1 and the solution Of the system (b) is x 1 =2, x 2 =1, x 3 = -1
Theorem Let 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 Let A and B be square matrices of the same size. If AB is invertible, then A and B must also be invertible
Theorem (Equivalent Statements) If A is an n×n matrix, then the following statements are equivalent A is invertible Ax = 0 has only the trivial solution The reduced row-echelon form of A is I n A is expressible as a product of elementary matrices Ax = b is consistent for every n×1 matrix b Ax = b has exactly one solution for every n×1 matrix b
A Fundamental Problem: Let A be a fixed mXn matrix. Find all mX1 matrices b such Such 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 has The unique solution x= b. If A is not square, or if A is a square but not invertible, then theorem does not Apply. In these cases the matrix b must satisfy certain conditions in order for Ax=b To be consistent.
Determine Consistency by Elimination Example: What conditions must b1, b2, and b3 satisfy in order for the system of equations To be consistent? Solution: The augumented matrix is Which can be reduced to row-echelon form as follows:
From the third row in the matrix that the system has a solution if and only if b 1, b 2, and b 3 satisfy the condition B 3 - b 1 - b 2 = 0 or b 3 = b 1 + b 2 R 2 -R 1 R 3 -2R 1 -R 2 R 3 +R 2
Example: What conditions must b1, b2, and b3 satisfy in order for the system of equations To be consistent? Solution: The augumented matrix is Which can be reduced to row-echelon form : In this case, there are no restrictions on b 1, b 2 and b 3. That is, the system Has the unique solution x 1 =-40b 1 +16b 2 +9b 3, x 2 =13b 1 -5b 2 -3b 3, x 3 =5b 1 -2b 2 -b 3