Using Augmented Matrices to Solve Systems of Linear …

1 Augmented Matrices - page 1 Using Augmented Matrices to Solve Systems of Linear Row OperationsTo Solve the Linear system algebraically, these steps could be + = = + = All of the following operations yield a system which is equivalent to the original. (Equivalent Systems have the same solution.) Interchange equations 2 and 3x5yz112x4y2z83z12+ = + = = Multiply equation 3 by 13x5yz112x4y2z8z1+ = + = = Multiply equation 2 by 12 x5yz11x2yz4z1+ = += = Add equation 1 to 2 and replace x5yz113y15z4+ = = = equation 2 with the result Multiply equation 2 by 13x5yz11y5z4+ = = = Multiply equation 2 by and add it 5 xz14y5z4 = = = to equation 1; replace equation 1 with the resultAdd equation 3 to equation 1; replace x18y5z4= = = equation 1 with the resultThe solution is (18,5,4).

2 Augmented Matrices - page that Produce Equivalent Systemsa) Two equations are ) An equation is multiplied by a nonzero ) A constant multiple of one equation is added to another matrix is a rectangular array of numbers written within brackets. The size of a matrix is always givenin terms of its number of rows and number of columns (in that order!). A 2 x 4 matrix has 2 rowsand 4 columns. Square Matrices have the same number of rows and columns. A matrix with a singlecolumn is called a column matrix, and a matrix with a single row is called a row matrix.

3 A squarematrix with all elements on the main diagonal equal to 1 and all other elements equal to 0 is called anidentity matrix. The 3x3 identity matrix is .100010001 The position of an element within a matrix is given by the row and column (in that order!) containingthe element. The element is in row 3 and column 4. 34a4. Elementary Row Operations that Produce Row-Equivalent Matricesa) Two rows are interchangedijRR b) A row is multiplied by a nonzero constantiikRR c) A constant multiple of one row is added to another rowjiikRRR+ (NOTE:means"replaces") an Augmented MatrixAn Augmented matrix is associated with each Linear system like x5yz113z122x4y2z8+ = = + = The matrix to the left of the bar is called the coefficient matrix.

4 15111003122428 an Augmented MatrixTo Solve a system Using an Augmented matrix, we must use elementary row operations to changethe coefficient matrix to an identity the Augmented matrix15111003122428 Interchange rows 2 and 315111242800312 23RR Augmented Matrices - page 3 Multiply row 3 by 131511124280014 331RR3 Multiply row 2 by 12 1511112140014 221RR2 Add row 1 to row 2 and replace 15111030150014 122 RRR+ row 2 with the result Multiply row 2 by 131511101050014 221RR3 Multiply row 2 by and add it to row 1.

5 5 1011401050014 2115 RRR + replace row 1 with the resultAdd row 3 to row 1; replace row 11001801050014 311 RRR+ with the resultThe solution is (18,5,4).