Matrices and Systems of Equations
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1 Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m n n n mn We will use the double subscript nottion for ech element of the mtri. For emple, the element of mtri A in the ith row nd the jth column is denoted b ij. The dimensions of mtri re given s rows columns. This is red m b n. A mtri is squre if it is n n; we s it hs order n. The min digonl of squre mtri is ll the elements ij for which i = j.
2 Emple: Determine the dimensions of ech mtri. CHAT Pre-Clculus Section 8. ) b) nswer: nswer: c) d) 8 8 nswer: nswer: A mtri tht hs onl one row is clled row mtri. A mtri tht hs onl one column is clled column mtri.
3 CHAT Pre-Clculus Section 8. Look t the sstem If we write the sstem of equtions without the vribles, ddition signs, nd equl signs, nd put it into mtri, we get wht is clled n ugmented mtri. or or (Different tetbooks will write them in vrious ws.) The coefficient mtri would be Note: An time term is missing from sstem of equtions, ero is put in its plce in the mtri.
4 CHAT Pre-Clculus Section 8. Elementr Row Opertions We need to trnslte the elementr row opertions from our lst unit into the lnguge of mtrices. Note tht these opertions will trnsform the ugmented mtri of sstem of equtions into the ugmented mtri of n equivlent sstem of equtions. If one mtri hs been obtined from nother mtri b using elementr row opertions, then the two mtrices re sid to row-equivlent. Here re the opertions. Elementr Row Opertions. Interchnge two rows.. Replce n row b nonero multiple of itself.. Replce n row b the sum of itself nd multiple of n other row in the mtri. We use the mtri version of the Gussin Elimintion method to solve sstems of liner equtions. Emple: Solve the sstem using the mtri version of Gussin Elimintion.
5 CHAT Pre-Clculus Section 8. Solution: Strt with the ugmented mtri. Multipl the st row b - nd dd to the nd row, replcing the nd row with our sum eqution. ) ( Multipl the st row b - nd dd to the rd row, replcing the rd row with our sum eqution. ) (
6 CHAT Pre-Clculus Section 8. Multipl the nd row b nd dd to the rd row, replcing the rd row with our sum eqution. () Replce the bottom row with - times itself. ) ( This finl mtri is sid to be in row-echelon form.
7 Row-Echelon Form nd Reduced Row-Echelon Form A mtri is in row-echelon form if it hs the following properties. CHAT Pre-Clculus Section 8.. All rows consisting entirel of eros occur t the bottom of the mtri.. The first nonero element of n row is, clled leding.. For two successive nonero rows, the leding in the upper row is frther to the left thn the leding in the lower row. A mtri is in reduced row-echelon form if the column with leding hs eros everwhere else in tht column. Emple: Continue working with the bove ugmented mtri until it is in reduced row-echelon form. Multipl the rd eqution b nd dd to the nd eqution, replcing the nd eqution with the sum. ()
8 CHAT Pre-Clculus Section 8. 8 Multipl the rd eqution b - nd dd to the st eqution, replcing the st eqution with the sum. ) ( Multipl the nd eqution b - nd dd to the st eqution, replcing the st eqution with the sum. ) ( The finl mtri is in reduced row-echelon form.
9 CHAT Pre-Clculus Section 8. Look t the row-echelon form nd the reduced row-echelon form. Use bck substitution to solve the sstems tht ech represents. First the row-echelon form: () () The solution is (-,, ). Now the reduced row-echelon form. The solution is (-,, ). **Both mtrices gve us the sme solution.
10 CHAT Pre-Clculus Section 8. Solving sstem of equtions b Gussin elimintion is tking the ugmented mtri of the sstem of equtions nd trnsforming it into row-echelon form, nd then using bcksubstitution to find the vlues of the vribles. Solving sstem of equtions b Guss-Jordn elimintion is tking the ugmented mtri of the sstem of equtions nd trnsforming it into reduced row-echelon form. The lst column gives the vlues of the vribles. Tip: Work column b column. Get the first column the w ou wnt it, then move to the second column, nd so on. Emple: Which of the following mtrices re in reduced row-echelon form? If it is not in reduced row-echelon form, stte whether it is in row-echelon form. ) b) 8 reduced row-echelon form reduced row-echelon form
11 CHAT Pre-Clculus Section 8. c) d) just row-echelon form reduced row-echelon form Emple: Solve the sstem b Gussin elimintion. 8 The solution is (, -, ).
12 CHAT Pre-Clculus Section 8. Emple: Solve b Guss-Jordn elimintion. The solution is (, -). Emple: Solve the sstem b Guss-Jordon elimintion. This sstem hs no solution.
13 CHAT Pre-Clculus Section 8. Emple: Solve the sstem b Guss-Jordon elimintion. 8 Note: With onl equtions, it is impossible to get rid of both nd in the top eqution. Let =. Then = - nd = 8, which mens = + 8 when we substitute for. The solution to the sstem is (+8, -, ). Emple: Solve the sstem. The solution is (-,, ).
14 CHAT Pre-Clculus Section 8. Mtri Opertions on the Grphing Clcultor Entering Mtri. Press [ nd ] [MATRX] [EDIT]. Choose [A] to enter our mtri s mtri A.. On the top of the edit screen ou must enter the dimensions (i.e. order) for our mtri. Move the cursor to ech spot nd enter the number of rows nd then columns.. Move the cursor down to the row, column entr spot. You will see the row, column reference t the bottom of the screen. Tpe in our entr. Press [ENTER]. Your cursor goes to the net element s spot.. Continue entering ll of the elements of the mtri. When finished, press [ nd ] [Quit].. To see the mtri on our min screen, press [ nd ][MATRX] [A] [ENTER]. Emple: Enter the ugmented mtri into our grphing clcultor s mtri A for the following sstem.
15 CHAT Pre-Clculus Section 8. Chnging Mtri to Row-Echelon Form. Press [ nd ] [MATRX] [MATH] [ref( ]. ( ref stnds for row-echelon form. ). On our min screen ou will see ref(. You need to tell the clcultor the nme of the mtri tht ou wnt it to put in row-echelon form. To do this, press [ nd ] [MATRX] [A] to enter mtri A. Press [ENTER]. Note: Row-echelon form mtrices re not unique. It depends on the sequence of row opertions chosen. Emple: Using the bove sstem nd mtri entered into our clcultor s mtri A, find the row-echelon form on our grphing clcultor Notice how it writes the frctions s decimls. To solve this sstem, we would hve to bck-substitute.
16 CHAT Pre-Clculus Section 8. Chnging Mtri to Reduced Row-Echelon Form. Press [ nd ] [MATRX] [MATH] [rref( ]. ( rref stnds for reduced row-echelon form. ). On our min screen ou will see rref(. You need to tell the clcultor the nme of the mtri tht ou wnt it to put in row-echelon form. To do this, press [ nd ] [MATRX] [A] to enter mtri A. Press [ENTER]. Note: Reduced Row-echelon form mtrices re unique. The will be the sme regrdless of the choices mde in row opertions. Emple: Using the bove sstem nd mtri entered into our clcultor s mtri A, find the reduced row-echelon form on our grphing clcultor. This tells us tht the solution to the sstem is (, -).
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