Module 16 - Matrix Analysis 1 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 16

MATRICES I

Module Topics 1. Terminology 2. Addition, subtraction and multiplication of matrices 3. Determinants

A:

Work Scheme based on JAMES (THIRD EDITION)

1. This module is the first of three on Matrices. Most of you will have met matrices at school/college, but a number of you will not have studied them at Advanced level and hence this module assumes no prior knowledge of the topic. Turn to p.256 of J. and read section 5.1. Matrices are an extremely important tool in solving large systems of simultaneous linear equations, and the latter arise in the solution of many engineering problems. Matrices also form the basis of many computer graphical packages, and occur in a variety of other situations. Next read section 5.2 on pp.258-260. Planes are considered in Module 15, but if you haven’t met them just accept that equations (5.2) represent four planes. 2. Study section 5.2.1. This section contains many definitions which must be known, so read it through more than once. Perhaps it should be emphasised that a unit matrix must be a square matrix, and then satisfy additional properties. Note that J. uses square brackets to represent a matrix, [1 2 3] for instance. It is more usual to use curved brackets, (1 2 3) , and curved brackets will be used here in the worked solutions, tests etc. Either notation will be accepted in your answers. 3. Having introduced the basic definitions you must now consider how to manipulate matrices. Study section 5.2.2 which defines equality, addition, multiplication by a scalar, properties of the transpose and the laws of addition. Note, in particular, that only matrices of the same order, i.e. with the same number of rows and the same number of columns, can be added or said to be equal. Study Example 5.1. A is a 3 × 3 matrix and B is a 3 × 2 matrix. These are different so cannot be added together (see part (a)). Similar reasons apply in parts (f ) and (i). Part (c) involves subtraction of matrices. Subtraction is defined in an analogous way to addition in that the two matrices must be of the same order (m × n say), and then elements in the resulting matrix, which will also be m × n, are found by subtracting the corresponding elements in the individual matrices. Another way of obtaining the same answer is to write C − A = C + (−A) = C + ((−1)A), and then use the results for multiplication by a scalar and for addition of matrices. 4. Study section 5.2.4 on matrix multiplication. The changes of axes discussed at the top of p.267 motivate the definition of matrix multiplication. The formal definition of multiplication is stated in the lower half of p.267, but it is probably easier to understand the definition from the diagram at the bottom of the page and the Example that follows it. The important points to emphasise are that the product AB exists only for matrices of certain orders (where the number of columns in A equals the number of rows in B) and, if the latter holds, then the element in the ith row and jth column of C (= AB) is formed from the –1–

ith row of A and jth column of B by adding together the products of the corresponding elements in the row and column (see the diagram at the bottom of p.267). If you have studied vectors, then you are essentially taking the scalar product of the ith row of A and the jth column of B, and hence this row and column must have the same length (i.e. have the same number of elements). Study carefully Examples 5.3 and 5.4. You should fill in the intermediate steps needed to get the final answers. If you encounter problems look again at part (a) of Example 5.3 and consider carefully which rows and columns are being multiplied together to give the individual elements in the product matrix. All the matrix products asked for in Examples 5.3 and 5.4 exist. This is not always the case, of course. In Example 5.3, for instance, A is 2 × 3 and b is 2 × 1, so you cannot calculate either Ab or bA. Read through the comments in the upper half of p.270. 5. Study the properties of multiplication, (a)-(f ), on pp.270 and 271. The fact that AB 6= BA, in almost all situations, is extremely important. Thus the order in which matrices are written in a product is crucial, and you must remember to retain the original order during any multiplicative operations, see properties (b), (c) and (d). Properties (e) and (f ) must also be known. Study Example 5.5. Standard software packages are available, but are not used in this Mathematics course. Our aims this year are that you understand the underlying mathematical techniques, and can work out simple examples. Note that in part (b) the equation BX = c has been solved by spotting a matrix A such that AB = I. Premultiplication of the original equation by A then removes B from the left-hand side, allowing X to be determined. This procedure will be investigated in more detail in Module 17. Work through Examples 5.6 and 5.7. ***Do Exercises 5, 6(a),(b) on p.279*** 6. Turn to p.282 and the important topic of determinants, which are defined only for square matrices. Study the theory at the beginning of section 5.3. The determinant of a 2 × 2 matrix and a 3 × 3 matrix are defined by expressions (5.9) and (5.10) respectively. You should observe that in the result (5.10) the elements in the 2 × 2 determinant multiplying the element a11 are the elements that remain in the original matrix B when the row and column that go through the element a11 are omitted. In a similar way the next two determinants are found in turn by omitting from B the row and column that pass through the element a12 , and then a13 . Study Example 5.12. ***Do Exercises 21(a),(b) on p.294*** 7. Study the theory in the bottom half of p.283. The general expression for evaluating the determinant of a n × n matrix A is equation (5.11), and you must be able to evaluate the minor and cofactor associated with any element of a matrix. Note that J. defines the minor and cofactor of any element in a determinant, but it is much more usual to associate them with an element in a matrix. The final result for evaluating the determinant of A, stated at the bottom of p.283, and the equivalent result in terms of columns, can be summarised as follows. You can calculate the determinant of any square matrix using any row or column. Choose a row (or column) of A, then |A| is determined by summing together the products of each element in that row (or column) with its corresponding cofactor. A 3 × 3 matrix has three rows and three columns and hence there are six ways of evaluating the determinant, in each case using just one of the rows or columns. Clearly, choosing the row or column in A with most zeros gives the least work. Study carefully Example 5.13 on p.284. To find M11 , the minor associated with the element a11 (which lies in the first row and first column of A), J. crosses out the first column and first row and then calculates the determinant of the resulting 2 × 2 matrix, which equals −2. Then, using the definition of cofactors, it follows that A11 = (−1)1+1 M11 = (−1)2 M11 = M11 = −2. In a similar way, J. shows that M12 = 2, from –2–

which A12 = (−1)1+2 M12 = (−1)3 M12 = −M12 = −2. The cofactor A13 is found in a similar way and the value of the determinant is then easily calculated by evaluating a11 A11 + a12 A12 + a13 A13 . ***Do Exercise 20 on p.294*** ***Do Exercise A: Evaluate the determinant of the matrix given in Exercise 21(c) on p.294 by expanding using the third row. 8. Determinants have a number of properties which can often shorten your calculations. Study the properties (a) to (g) discussed on pp.284-286, but the proofs need not be remembered. Properties (f ) and (g) will be used in later situations. On the other hand, property (d) is not normally listed, and the others ((a), (b), (c), (e)) are sometimes helpful to simplify calculations but are not essential in order to obtain answers. Work through Examples 5.14 and 5.15. Both Examples illustrate some of the properties of determinants. Note that the answers to parts (a), (b) and (c) of Example 5.15 are consistent with the result |AB| = |A| |B|, i.e. 0 = 0(2). Read the paragraph just below Example 5.15, and work through Example 5.16. Then read the first paragraph only underneath the last Example, just below the horizontal line. 9. Move to p.290 and study the two lines in the lower half of the page below the horizontal line, including expression (5.14) which defines the adjoint matrix. As you will see in Module 17 the adjoint matrix appears in a well-known formula for calculating the inverse of a matrix. Move on to p.292, and work through Example 5.19. ***Do Exercises 28(b), 23 on p.294***

B:

Work Scheme based on STROUD (FIFTH EDITION)

Most of the material in this Module is in S. The initial work on matrices starts in Programme 5 on p.536. Read frames 1-23. Next turn to Programme 4 on determinants starting on p.502. Work through frames 1-7 on second order determinants, or determinants of 2 × 2 matrices. Then study frames 17-27 on third order determinants, which starts on p.509, and frames 59-66 on properties of determinants, beginning on p.526. Return to the Programme on matrices and start at p.546. Work through frames 24-27, on cofactors and the adjoint matrix. You will now have covered almost all of the material in this module. However, S. does not state some of the properties of matrix multiplication and, if you wish to fill in this gap, you should turn to J. and section 5 of the Work Scheme above and study pp.270 and 271.

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Specimen Test 16 1.

If

 A=

1 3

2 1

−1 4



 ,

B=

−2 1 3 2



 ,

C=

0 −1

−2 1

 3 1 D =  2 −1  , 0 1 

 ,

write down the order (in the form m × n) of each of the above matrices. State which of the following exist and express as a single matrix those which do: (ii) AT − DT ,

(i) 2C − B,

(iv) ABT ,

(iii) AD,

(v) DC.

2.

State the condition for a matrix A to be symmetric.

3.

For general matrices express each of the following in equivalent terms e.g. (ABT )T = (BT )T AT = BAT (i) (AT )T ,

4.

(ii) (AT B)T .

Assuming the products between the matrices P, Q, R exist, are the following relations always satisfied? (i) (P + Q)R = PR + QR,

5.

In the determinant

(ii) PQ = QP.

5 2 −7 6 11 4 3 8 1

evaluate (i) the minor of the element in the first row and third column, (ii) the cofactor of the element in the second row and second column.

6.

Evaluate directly the determinant in question 5 by expanding in terms of elements of the third column. 

7.

If

1 A= 3 −2

 2 1 0 −1  1 1

determine adj A.

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