Introduction To Matrices For Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester University1What is a Matrix?A matrix is a rectangular array of elements, usually numbers, e.g. 1 -2 4123800 02 -1 117The above matrix is a (4 3)–matrix, i.e. it has three columns and four rows.1.1Why use Matrices?We use matrices in mathematics and engineering because often we need to deal with severalvariables at once—eg the coordinates of a point in the plane are written (x, y) or in space as(x, y, z) and these are often written as column matrices in the form: xy and x y zIt turns out that many operations that are needed to be performed on coordinates of points arelinear operations and so can be organized in terms of rectangular arrays of numbers, matrices.Then we find that matrices themselves can under certain conditions be added, subtracted andmultiplied so that there arises a whole new set of algebraic rules for their manipulation.In general, an (n m)–matrix A looks like: a11a21a31. A an 1,1an,1a12a22a32.a13a23a33.an 1,2an,2an 1,3an,3.a1,m 1a2,m 1a3,m 1.a1,ma2,ma3,m.an 1,m 1an,m 1an 1,man,m Here, the entries are denoted aij ; e.g. in Example 4.1.1 we have a11 1, a23 2, etc. Capitalletters are usually used for the matrix itself.1.2DimensionIn the above matrix A, the numbers n and m are called the dimensions of A. Basedon original lecture notes on matrices by Lewis Pirnie1
21.3Introduction to MatricesAdditionIt is possible to add two matrices together, but only if they have the same dimensions. In whichcase we simply add the corresponding entries: 1 -2 4123800 40 02 -1 5111720-811 15 -2-3 -2 91-503011 1-1 -3 67If two matrices don’t have the same size (dimensions) then they can’t be added, or we say the sumis ‘not defined’.1.4Example 1 26 417 74 0 -2 12 1 0 3 7003 7 1 24 0 0 -2 is undefined3 -23 7 22.1Multiplying MatricesWhy should we want to?We can motivate the process by looking at rotations of points in the plane. Consider the point Pon the x-axis at x x, y 0, so it has coordinates x0Now, let P’ be the point obtained by rotating P round the origin in an anticlockwise directionthrough angle θ, keeping the distance from the origin (which is actually the square root of the sumof squares of the coordinates) constant. What are the coordinates of the point P’ ?It is not difficult trigonometry to work out that these are: x cos θx sin θNext consider the point Q on the y-axis with coordinates 0yand rotate this point round the origin in an anticlockwise direction through angle θ, keeping thedistance from the origin constant. We find that the new coordinates are: y sin θy cos θNow the payoff, it is actually a 2 2 matrix that does the rotating here because the operation islinear on the coordinate components. We can express the rotation of any point with coordinates(x, y) as the following matrix equation: cos θ sin θxx cos θ y sin θ sin θcos θyx sin θ y cos θCan you guess what will be the matrix for rotation in a clockwise direction?
3C.T.J. Dodson2.2Exercise on Trigonometry1. Three common right angled triangles can be used to obtain the following values for thetrigonometric functions:30 π6sin π6 cos π6 tan π6 45 π4sin π4 12 32 1360 π3sin π3 12 12cos π4 tan π4 1cos π3 tan π3 3212 32. Obtain the rotation matrices explicitly for rotations of θ 30 , 45 , 60 , 90 , 180 .2.3RulesWhen multiplying matrices, keep the following in mind: lay the first row of the first matrix on topof the first column of the second matrix; only if they are both of the same size can you proceed.The rule for multiplying is: go across the first matrix, and down the second matrix, multiplyingthe corresponding entries, and adding the products. This new number goes in the new matrix inposition of the row of the first matrix, and the column of the second matrix. For example: 1324 210537 4101.2 2.13.2 4.110517371.0 2.53.0 4.51.3 2.73.3 4.7 Symbolically, if we have the matrices A and B as follows: a11a21a31. A an 1,1an,1 b11b21b31. B bp 1,1bp,1a1,ma2,ma3,m. a12a22a32.a13a23a33.an 1,2an,2an 1,3an,3.b12b22b32.b13b23b33.b1,qb2,qb3,q.bp 1,2bp,2bp 1,3bp,3bp 1,q 1.bp 1,qbp,qan 1,man,m , then the product AB is given by: Pma1i bi1Pi 1m a2i bi1i 1 .Pm .i 1 ani bi2Pma1i bi2Pi 1mi 1 a2i bi2Pm . . .i 1 ani bi2.Pma1i biqPi 1mi 1 a2i biqPm . . .i 1 ani biq Pmwhere i 1 a1i bi1 stands for a11 b11 a12 b21 a13 b31 . . . a1n bn1 , etc.Note that we must have m p, i.e. the number os columns in the first matrix must equal thenumber of rows of the second; otherwise, we say the product is undefined.
42.4Introduction to MatricesExampleWe multiply the following matrices: 1 3-1 2 0 1.1 2.0 0.7 1.3 2. 1 0.5-1 -5 0 -1(i) 4 1 -24.1 1.0 2.7 4.3 1. 1 2.5-10 17 5 4 72 0 -8 1 is undefined(ii)23 -12 1 17 2 24.2 12 . 1 4.2 12 .44 1262 (iii) -1 4212.2 1. 12.2 1.43 8 -1 10 -1 2 1(iv) 0 2 3 7 0 -11 4 1.0 1.3 1. 1 1.7 1.2 1.0 1.1 1. 138 -2 -20. 1 2.70.2 2.00.1 2. 1 6 14 0 -2 0.0 2.31.0 4.31. 1 4.71.2 4.01.1 4. 112 27 2 -3 1 021.2 0.52(v) 3 153.2 1.511 4 01 61 3 0 0 -8 -3 0 (vi)-2 8 25 1 -2 1 1.4 3.0 0.51.0 3. 8 0.11.1 3. 3 0. 21.6 3.0 0.1 2.4 8.0 2.5 2.0 8. 8 2.1 2.1 8. 3 2. 2 2.6 8.0 2.1 4 -24 -86 2 -62 -30 -10We see from the roductProductofofofofof(2 3)–matrix(2 2)–matrix(3 2)–matrix(2 2)–matrix(2 3)–matrixwithwithwithwithwithaaaaa(3 2)–matrix(2 2)–matrix(2 4)–matrix(2 1)–matrix(3 4)–matrixisisisisisaaaaa(2 2)–matrix.(2 2)–matrix.(3 4)–matrix.(2 1)–matrix.(2 4)–matrix.Note that the two middle numbers must be the same if the product is defined; and then thedimensions of the answer is just the two outer numbers. Thus, the product of an (n m)–matrixwith a (m q)–matrix is an (n q)–matrix.3Scalar MultiplicationThere is another type of multiplication involving matrices called scalar multiplication. This meansjust multiplying each entry of the matrix by a number. For example: -1 2 0-3 6 03 4 1 -212 3 -63.1RulesThere are some rules which matrix addition and multiplication obey:AssociativeCommutative(A B) C A (B C)A B B A
5C.T.J. mutativeMoving ConstantsA(B C) AB AC(A B)C AC BC(AB)C A(BC)AB 6 BA (usually)A(λB) λ(AB)(Assuming that the sums and products are defined in all cases.)3.2Example Let A Consider the following: 1 0-1AB 3 21 BA -114-2 13 02-11, B 4-2 02 13 4-2 23, C 1. 1 0.13. 1 2.11.4 0. 23.4 2. 2 1.1 4.31.1 2.3 1.0 4.21.0 2.2 -1-111-548 8-4 Now, AB 6 BA, and we see that two matrices are not the same if they are multiplied the otherway around. Also consider: 1 022AC 3 2312 BC -11 212 02 04AB BC A B 13(A B)C 4-210-4 10-4 128 -114-2 0440 23128 40 23 and we notice that (A B)C AC BC as required.4TransposeAnother operation on matrices is the transpose. This just reverses the rows and columns, orequivalently, reflects the matrix along the leading diagonal. The transpose of A is normally writtenAt thus a11a12a13.a1,m 1a1,m a21a22a23.a2,m 1a2,m a31a32a33.a3,m 1a3,m ,A . . an 1,1 an 1,2 an 1,3 . . . an 1,m 1 an 1,m an,1an,2an,3.an,m 1an,m
6Introduction to Matrices a11a12a13. At a1,n 1a1,na21a22a23.a31a32a33.a2,n 1a2,na3,n 1a3,n. am 1,1am 1,2am 1,3.am,1am,2am,3.am 1,n 1am 1,nam,n 1am,n Note that the transpose of a (n m)–matrix is a (m n)–matrix.4.1ExampleAs an example of the transpose: A 52043-15 2, At 4-1 03 5Square MatricesGiven a number n, (n n)–matrices have very special properties. Note that if we have two(n n)–matrices, the product is defined and will also be an (n n)–matrix.5.1The Identity MatrixThere also exists a special matrix, known as the identity, In n : I2 2 1001 , I3 31 00010 10 00 , I4 4 01001000010 00 , etc.0 1which has 1’s on the main diagonal, and 0’s everywhere else. This matrix has the property that,given any (n n)–matrix A:AIn n A and In n A Ai.e. multiplying by the identity on either side doesn’t change the matrix. This is similar to theproperty of 1 when multiplying numbers. We usually abbreviate In n to just I when it’s obviouswhat n is.5.2Example Let A 1372 , I I2 2 1001 Now, we check the properties of the identity: AI IA as required.1372 1001 1001 1372 1.1 7.03.1 2.01.0 7.13.0 2.1 1.1 0.30.1 1.31.7 0.20.7 1.2 1372 1372
7C.T.J. Dodson6DeterminantsOne of the most important properties of square matrices is the determinant. This is a numberobtained from the entries.6.1Determinant of a (2 2)–Matrix Let A 6.2acbd . Then, the determinant of A, denoted det A or A is given by ad bc.Example det det-13212-635 2.5 3.1 10 3 7 ( 1)( 6) 2.3 6 6 0Before we go on to larger matrices, we need to define minors.6.3MinorsLet A be the (n n)–matrix a11a21a31. A an 1,1an,1a12a22a32.a13a23a33.an 1,2an,2an 1,3an,3. a1,m 1a2,m 1a3,m 1.a1,ma2,ma3,m.an 1,m 1an,m 1an 1,man,m Then, the minor mij , for each i, j, is the determinant of the (n 1 n 1)–matrix obtained bydeleting the ith row and the j th column. For example, in the above notation: a22a23.a2,m 1a2,m a32a33.a3,m 1a3,m .m11 det an 1,2 an 1,3 . . . an 1,m 1 an 1,m an,2an,3.an,m 1an,m m216.4a12a32. det an 1,2an,2a13a33.an 1,3an,32We compute all the minors of A 0-5m21 a1,m 1a3,m 1.a1,ma3,m.an 1,m 1an,m 1an 1,man,m Example m11 .40103-2-1-2 8m12 2m22 140 -13 -20-52-53-2-1-2 15m13 9m23 0-52-54010 20 5
8Introduction to Matricesm31 6.514-13 720m32 -13 6m33 2014 8Minors and CofactorsThe numbers called ‘cofactors’ are almost the same as minors, except some have a minus sign inaccordance with the following pattern: . . . . The best way to remember this is as an ‘alternating’ or ‘chessboard’ pattern. The cofactors fromthe previous example are:c11 m11 8c21 m21 2c31 m31 77c12 m12 15c22 m22 9c32 m32 6c13 m13 20c23 m23 5c33 m33 8Determinant of a (3 3)–MatrixIn order to calculate the determinant of a (3 3)–matrix, choose any row or column. Then, multiplyeach entry by its corresponding cofactor, and add the three products. This gives the determinant.7.1Example 2Letting A 0-5 -13 as before, we compute the determinant using the top row:-2140det A a11 c11 a12 c12 a13 c13 2.( 8) 1.( 15) ( 1).20 16 15 20 51Suppose, we use the second column instead:det A a12 c12 a22 c22 a32 c32 1.( 15) 4.( 9) 0.( 6) 15 36 0 51It doesn’t matter which row or column is used, but the top row is normal. Note that it is notnecessary to work out all the minors (or cofactors), just three.7.2Example 1Let B -23012 40 . We compute the determinant of B:1 1det -23012 40 111201 0-2301 4-23 1(1 0) 0( 2 0) 4( 4 3) 1 28 2712
9C.T.J. Dodson8Determinant of an (n n)–MatrixThe procedure for larger matrices is exactly the same as for a (3 3)–matrix: choose a row orcolumn, multiply the entry by the corresponding cofactor, and add them up. But of course eachminor is itself the determinant of an (n 1 n 1)–matrix, so for example, in a (4 4) determinant,it is first necessary to do four (3 3) determinants — quite a lot of work!9InversesLet A be an n n matrix, and let I be the n n identity matrix. Sometimes, there exists a matrixA 1 (called the inverse of A) with the property:A A 1 I A 1 AIn this section, we demonstrate a method for finding inverses.9.1Inverse of a 2 2 Matrix Let A ac . Then, the inverse of A, A 1 is given by:bdA 1 1ad bc d c ba To check, we multiply:A 1 A 1ad bc d c ba 1ad bc acad bc0bd 1da bc ca acad bc 01 0 ad bc0 1 In a similar fashion we could show that A A 1 I.Of course, the inverse could also be written 1d 1A cdet A badb bd cb ad Note, that if det A 0, then we have a division by zero, which we can’t do. In this situation thereis no inverse of A.9.2Inverse of 3 3 (and higher) MatricesRecall the definition of a minor from Section 4.3.3: given an (n n)–matrix A, the minor mij isthe determinant of the (n 1 n 1)–matrix obtained by omitting the ith row and the j th column.9.3Example 1Let A -23m11 m21 m31 120201014140012 40 . We calculate the minors:1 1m12 8m22 4m32 -2 0 23 11 4 113 11 4 8-2 0m13 m23 m33 -2 1 73 21 0 23 21 0 1-2 1
10Introduction to MatricesRecall also the pattern of and signs from which we obtain the cofactors: Now, we put the minors into a matrix and change their signs according to the pattern to get thematrix of cofactors: 12 7 8 11 2 4 81The next stage is take the transpose: 1 2 78 11 2 4 8 1and finally we must divide by the determinant, which is 27, from Example 4.3.8: 841 27 2718 42712118 2 11 8 27A 1 2727 27712 7 21 272727This shows how to calculate the inverse of a (3 3)–matrix. We check the result: 18 41 0 41 16 120 8 81 11 8 -2 1 0 2 22 24 0 11 16A 1 A 227 7 213 2 1 7 4 30 2 2 -27001 0 01-270 0 1 0 02700-270 0 1 4 0 48 0 8 28 0 1as required.The same procedure works for (n n)–matrices.(I)(II)(III)(IV)Work out the minors.Put in the signs to form the cofactors.Take the transpose.Divide by the determinant.Furthermore, an n n matrix has an inverse if and only if the determinant is not zero. So, it’sa good idea to calculate the determinant first, just to check whether the rest of the procedure isnecessary.10Linear SystemsWe discuss one very important application of finding inverses of matrices.10.1Simultaneous EquationsOften, when solving problems in mathematics, we need to solve simultaneous equations, e.g. something like:2x5x y3y 37
11C.T.J. Dodsonfrom which we would obtain x 2 and y 1. The process we have used up until now is a littlemessy: we combine the equations to try and eliminate one of the unknown variables. There is amore systematic way using matrices. We can write the equations in a slightly different way: 2x y3 5x 3y7Now we can check that the first matrix is equal to the product: 2x y2 1x 5x 3y5 3yand so altogether we have a matrix equation: 2 1x3 5 3y7The next stage is to use the inverse of the (2 2)–matrix, so let’s calculate that now. Let A 25 , then A 1 1312.3 1.5 3-5-12 -12 3-537 -12 .Now, we take the matrix equation above, and multiply by A 1 2 1x3 5 3y7 3-5-12 2513 xy 3-5Then, doing the multiplication: 1 0x3.2 1.72 0 1y 5.3 2.7 1 xy 2 1 and so x 2 and y 1, as required. So, provided we can work out the inverse of the matrix ofcoefficients, we can solve simultaneous equations.11Larger SystemsThe same thing works with 3 equations in x, y and z. Suppose we havex 2x 2y3yy 2z2z8z 127Then, the matrix form is 1 0223-1 2x-1-2 y 2 8z7Now, we denote the 3 3 matrix by A, and calculate the inverse of A. The minors are as follows:
12m11 m21 m31 Introduction to Matrices3 -2 22-1 82 2 18-1 82 2 103 -2021210m12 m22 m32 -2 482 482 2-2m13 m23 m33 0212103 6-12 5-12 33Recall the chessboard pattern: So we have the following matrix of cofactors: 22 18 10 442 65 3We can calculate the determinant by taking any row or column, and multiplying the original matrixentry by its corresponding cofactor, and then adding — let’s choose the top row:det A 1.22 2. 4 2. 6 22 8 12 2and so the determinant is 2, and so we will be able to find the inverse. From the matrix of cofactors,we take the transpose, and then divide by the determinant to get A 1 : 11 9 522 18 10121 42 2A 1 4253 653 322Now, we return to solving the simultaneous equations, where x1 22 0 3 -2 y z2 -1 8we had: -12 7Multiplying both sides on the left by A 1 , we have: 11 9 511x1 22 221 0 3 -2 y 2532 -1 8z 3 322 9252 5-11 2 372and we know that A 1 A I, so 11. 1 9.2 5.71 0 0x 0 1 0 y 2. 1 2.2 1.7 0 0 1z 3. 1 25 .2 32 .7 64x y 13 37z2and so we get x 64, y 13 and z 372 .It’s a very good idea to check calculations like this!x 2y 2z 64 2(13) 2( 372 ) 64 26 373y 2z 3(13) 2( 37) 39 3722x y 8z 2( 64) 13 8( 372 ) 128 13 148as required. 1 2 7
C.T.J. Dodson1213Appendix12.1Scientific Wordprocessing with LATEXThis pdf document with its hyperlinks was created using LATEX which is the standard (free) mathematical wordprocessing package; more information can be found via the webpage [1].12.2Computer Algebra MethodsThe computer algebra package Mathematica [2] can be used to manipulate and invert matrices.Similarly, Maple and Matlab also can be used for working with matrices.References[1] On-line mathematical and.ac.uk/ history/Mathematicians/Alfred Gray’s Mathematica NoteBooks on differential ks/3759Elementary Notes on:Curves http://www.maths.manchester.ac.uk/ kd/curves/curves.pdfSurfaces http://www.maths.manchester.ac.uk/ kd/curves/surfaces.pdfKnots http://www.maths.manchester.ac.uk/ kd/curves/knots.pdfLaTeX Tutorial:http://www.maths.manchester.ac.uk/ kd/latextut/pdfbyex.htm[2] S. Wolfram. The Mathematica Book Cambridge University Press, Cambridge 1996.
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester University 1 What is a Matrix? A matrix is a rectangular array of elements, usually numbers, e.g. 0 B B @ 1 3 0-2 8 2 4 0 -1 1 2 0 117 1 C C A The above matrix is a (4 3){matrix, i.e. it has three columns and four rows. 1.1 Why use Matrices?
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Class XII – NCERT – Maths Chapter 3 - Matrices 3.Matrices . Exercise 3.1 . Question 1: In the matrix 2 5 19 7 5 35 2 12 2 3 1 5 17. A . As the given matrices are equal, their corresponding elements are also equal. Class XII – NCERT – Maths . Chapter 3 - Matrices . 3.Matrices . Comparing the corresponding elements, we get: .
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