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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

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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.

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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

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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 | ]

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Row operations rref Thus

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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.

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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.

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Example: Solve the system by using Let We showed in previous section that Then Thus x 1 = 1, x 2 = -1, x 3 =2

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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)

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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

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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

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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

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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.

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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:

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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

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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

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