Year 11 Matrices Semester 2. Yuk

Transcription

1 Year 11 Matrices Semester 2 Chapter 5A input/output Yuk 1

2 Chapter 5B Gaussian Elimination an Systems of Linear Equations This is an extension of solving simultaneous equations. What oes a System of Linear Equations mean? Recall that Linear means straight. You woul remember solving simultaneous equations in Maths B is like fining the intersection of 2 straight lines. System - It s a bit like; a couple is two an a few is three well in Maths, as soon as you have three equations, we say it is a System of Linear Equations. We are simply looking to fin the point of intersection of a number of straight lines, an where there are three variables (means we get a line in 3D), we nee three equations. Simply, we nee one point that satisfies all three equations, the point of intersection of three straight lines in 3 Dimensions! Think of how easy it is to make up 2 linear equations in 2D that intersect it s not so easy in 3D!!! Consier the System: 2x + 3y 5z = 8 x + 5y + 7z = 3 6x + 2y = 1 can be written in the form AX = B x y z = Clearly it is VERY important to line everything up in vertical columns PRIOR to putting your equations into the Augmente matrix. ** note the zero in position a!,! ** You coul fin x, y, z by pre-multiplying both sies by the Inverse Matrix, however as matrices get bigger, the process of fining the Inverse Matrix becomes more teious So, lets try Gaussian Elimination J 2

3 Gaussian Elimination Gaussian elimination is applies similar techniques as the elimination process in Maths B, however this is MATHS C, so you MUST comply carefully with the various matrix forms use in these solutions as your exams are going to a MATHS C PANEL. First we nee to put the set of equations into an Augmente Matrix: Augmente Matrix: An augmente matrix is combining two separate matrices into a single matrix form. (Augmente: having been mae greater in size, like an augmente pension if you are over 65; or; containing an interval which is one semitone greater than the corresponing major or perfect interval) *** check you can see how we get from AX = B, to the augmente matrix. *** *** (this is an Example only o not try an solve this one) *** Then procee to manipulate with Permitte Operations: 1. Multiply a Row by a scalar 2. A or subtract one row from another 3. Swap the positions of two rows (we o these in Maths B alreay, but now we just have to get use to Matrix setting out) We are looking for manipulation to get the Augmente Matrix into Reuce Row Echelon Form (example of Reuce Row Echelon form) Reuce Row Echelon Form nees to have the Ientity matrix as the leaing part of the augmente matrix. In the above example, Reuce Row Echelon Form prouces a simple solution to an Augmente matrix. Here, x = 3, y = 5 an z = 2 How o we get that? lets take a closer look 3

4 Take this augmente matrix; We can now pull the Augmente Matrix back apart an re-convert it to a system of three linear equations. Technically, we get x = 3, from the First Row taking back from Augmente matrix form into equations form, Row 1 becomes 1x + 0y + 0z = 3 similarly from Row 2 we get 0x + 1y + 0z = 5 an 0x + 0y + 1z = 2 Whalla x = 3, y = 5 an z = 2 Although you may see a short cut metho here, o NOT take any short cuts. Use legal row operations to get the Augmente matrix all the way to Reuce Row Echelon Form, an then present your solution irectly from this. When performing Gaussian Elimination operations, label clearly what you o on each step an o one step per line. (similar to how you labelle your elimination steps in Maths B) 4

5 We can also use the Gaussian metho to fin an Inverse Matrix. We covere Inverses in Term 1, an this is an Alternative metho that obtains the same result. You woul recall: Given Matrix A = a b c, then A!! =!!"!!" c b a We o a chapter on the Inverse Matrix in a couple of lessons that uses a ifferent technique, but as we are in Maths C, we like to o things many ifferent ways. This is a nice easy way, so lets take a look: To fin A!! We simply input to an Augmente matrix A I an use the Gaussian techniques until we get the Ientity matrix at the front, an we en up with: I A!! Cool hey! Fining Inverse Matrices by the Gaussian metho will be a straight forwar process once you get use to the Row Operations. 5

6 Chapter 5C Determinants Recall that the Determinant of a matrix is a single value! Given Matrix A = a c b The Determinant of A is written as: A = For a 3X3 Matrix: a b c = a bc A = a b c e f g h i = a e f h i b g f i + c e g h = a ei fh b i fg + c(h eg) Note: you can expan ANY row or column I suggest you choose wisely! OK i you notice that you SUBTRACT the b(i-fg), but this is NOT always the case. How o you know when it s a NEGATIVE, or a POSITIVE? There is a pattern, as per the text book page 203, but I prefer you think back to our general form of a Matrix a!! a!" a!" a!" a!! a!" a!" a!" a!! If the sum of the Row an Column subscripts are Even, then its Positive, if its O then its Negative! This process looks harer than it actually is. Take a look at my vieo if you nee. 6

7 Chapter 5D More on Determinants I won t focus too much on tricks, but certainly look to Short-cut your Matrix Determinant calculations if you can: - choose a row with zero s (or small numbers) in it to make calculations Quicker Choose your row carefully: Rather than expan Row 1, lets choose Column 2 as it simplifies the calculations = = = 44 Other Tricks: Page 201 in reality these on t come along too often. It is MORE important that you can Calculate the eterminant, than remember the following tricks - Determinant of a transpose.not too sure when this woul happen in reality? - Ientical Rows et A = 0 - row or column all zero s et A = 0 - interchanging rows not something that I see value in examining - multiples of rows - aing rows/columns - if one row is a scalar multiple of another row, then et A = 0 - Upper Triangular Rule says that if there are zero s in all spaces above the iagonal, then the Determinant is simply the prouct of the iagonals - there is a Lower triangular rule also Lower Triangle rule: Upper Triangle rule: = 30 = 10 *** Do NOT o anything on 4X4 matrices! Nope, not even for question 9 or 10. *** 7

8 Chapter 5E Inverse of a Matrix You woul recall: Given Matrix A = a b c, then A!! =!!"!!" c b a We learnt this by rote, but for bigger Matrices, we will use a Rule: A!! =!"#$! For ANY size Matrix, if the Determinant is 0, then the matrix is Singular an oes NOT have an inverse! So, what is the Ajoint or Ajugate Matrix (aja). First we nee to know about the Cofactor Matrix: If, A = a b c e f g h i, then A!"#$%&"' = e h b h b e f i c i c f g a g a f i c i c f g a g a e h b h b e ** you will notice that pattern again with the negatives ** Once you have the Cofactor Matrix we Transpose it to get aja, so aja = A!! *** Recall, the Text book enotes the Transpose matrix as A!, but I prefer A!. *** An we know how to calculate the Determinant J So, again, where, A!! = aja A aja = A!! Strategy? o it in steps 1. Calculate the eterminant 2. fin the cofactor matrix 3. convert to ajoint matrix 4. then scalar multiply by!!"#"$%&'('# 8

9 Chapter 5F Cramer s Rule As with ifferent areas of maths, there are a number of ways to arrive at the answer. Cramer s rule is just another tool to have with which you may solve problems. It looks way more complicate than it is. Once you see how these matrices go together, it is quite straight forwar J Given a set of linear equations: a 1 x + b 1 y + c 1 z = 1 a 2 x + b 2 y + c 2 z = 2 a 3 x + b 3 y + c 3 z = 3 convert to the form Ax = b " a 1 b 1 c 1 % " x% " $ ' $ ' $ $ a 2 b 2 c 2 ' $ y' = $ # $ a 3 b 3 c 3 &' # $ z& ' # $ % ' ' &' here the values of x,y an z can be foun as per: x = A 1 A y = A 2 A z = A 3 A where A = a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 an 1 b 1 c 1 a 1 1 c 1 a 1 b 1 1 A 1 = 2 b 2 c 2 A 2 = a 2 2 c 2 A 3 = a 2 b b 3 c 3 a 3 3 c 3 a 3 b 3 3 This is one where you just nee to REMEMBER the process. There are ways to make it easier to remember, but I m not too sure we want to try an fin out how & why it all works? 9

10 Set of linear equations: 3x + 5y = 1 x 2y = 4 Task 1. Put this simple set of linear equations into the form Ax = B Pre-multiply both sies by A!! to solve for the unknowns. Task 2. Put the above set of linear equations into an augmente matrix an use row operations to reuce to echelon form, an hence solve for the unknowns. Task 3. Take your previous augmente matrix, an work through to Reuce Row Echelon Form an solve for x an y. Task 4. Use Cramer s rule to solve the equations. Task 5. Ensure your answers in the first 4 tasks are the same! Lets move on to bigger matrices! 10

11 Solve the following systems of linear equations in Every way you can...!!! x + 3y + z = 10 x 2y z = 6 2x + y + 2z = 10 (1, 2 & 3) more practice? x + 2y + 2z = 5 3x 2y + z = 6 2x + y z = 1 (-1. 2 & 1) more practice? x + 2z = 9 2y + z = 8 4x 3y = 2 (1, 2 & 4) 11