Augmented Matrices - page 1Using Augmented Matrices to Solve Systems of Linear Equations1.Elementary Row OperationsTo solve the linear system algebraically, these steps could be used.x5yz113z122x4y2z8+−=−=+−=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).−Augmented Matrices - page 22. Operations that Produce Equivalent Systemsa) Two equations are interchanged.b) An equation is multiplied by a nonzero constant.c) A constant multiple of one equation is added to another equation.3.MatricesA 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. 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")→5.Forming 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. 15111003122428−−−6. Solving an Augmented MatrixTo solve a system using an augmented matrix, we must use elementary row operations to changethe coefficient matrix to an identity matrix.Form the augmented matrix15111003122428−−−Interchange rows 2 and 315111242800312−−−23RR↔Augmented Matrices - page 3Multiply row 3 by 131511124280014−−−331RR3→Multiply row 2 by 12−1511112140014−−−−−221RR2−→Add row 1 to row 2 and replace 15111030150014−−−122RRR+→row 2 with the result Multiply row 2 by 131511101050014−−−221RR3→Multiply row 2 by and add it to row 1;5−1011401050014−−2115RRR−+→replace row 1 with the resultAdd row 3 to row 1; replace row 11001801050014−311RRR+→with the resultThe solution is