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Row and Reduced Row Echelon Elementary Matrices

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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.

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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.

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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.

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

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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.

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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.

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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)

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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.

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Using Elementary Matrices

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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.

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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.

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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

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

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E2 = E2A = = AE2 = =

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E3 = E3A = = AE3 = =

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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.

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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

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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

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Compute the inverse of A for A =

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Now, solve the system:

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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

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Keys to calculating Inverses

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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

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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

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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

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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

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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.

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

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Using EROs on rows 2 and 3

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The matrix will be row equivalent to I iff:

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This implies that the Det(A) =

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Use EROs to find:

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

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

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Factorize the determinants of What is ?

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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:

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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: =

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Now, the matrix corresponds to Since =

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Then =

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

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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.

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

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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.

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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.

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

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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.

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

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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”

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

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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.

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It is easy to show that

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

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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

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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

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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.

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

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Use Cramer’s rule to write down the solution to the system

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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.

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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

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

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Taking the determinant Implying that and therefore; exists

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Since, Multiplying through by A

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Multiplying through by A -1

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Since

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