MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2

Transcription

1 MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous equations. Calculate determinants. Calculate minors and cofactors. Define and form the adjoint matrix. Define and form the inverse matrix. Define the augmented matrix. Use all the above to solve simultaneous equations.. INTODUCTION In tutorial on matrices you were introduced to some of the basic terms and operations that we can do with them. In this tutorial we shall look at more advanced ideas and how they are used to solve simultaneous equations. D.J. Dunn

2 . OUTLINE TO A of SIMULTANEOUS EQUATIONS WITH A MATIX An important use of matrix theory is the solution of problems with more than two unknown variables and because the method is based on strict rules, it is suitable for use in computer programmes. Consider how three simultaneous equations are presented as a vector. The Matrix A is called the COEFFICIENT MATIX and it is formed from an array of numbers made from the coefficients a a The column vector b is made from the coefficients b, b and the column Vector X is made up from the variables x, y and z. The column vector b is the product of matrix A and X. a x + a y + a z = b a x + a y + a z = b a x + a y + a z = b We write this as A X = b Suppose that we had a matrix B such that AB = I (The unit matrix). We could then state : ABX = I X = B b X = B b This gives a numerical solution for X. The problem is finding the matrix B such that AB = I. This is a matrix called the inverse matrix and we must understand the following work in order to find it.. DETEMINANTS The determinant of a matrix is a single number that results from performing a specific operation on the array. It will be used later to solve simultaneous equations. The determinant of a matrix A is denoted as det A or A. The rule for finding the determinant can only be applied to a square matrix and the following is an explanation of it. For a single element array the determinant is the element. A = [a ] A = a For a x array the determinant is found as follows. A = (a a ) - (a a ) WOKED EXAMPLE No. Calculate A A = ()(-) ()(4) = -4 - = -6 WOKED EXAMPLE No. Find the determinant of a x unit matrix. A = ()() ()() = D.J. Dunn

3 An important point to remember is that the determinant of all unit matrices is. For larger square arrays, the rule for finding the determinant is more complicated and it is crucial to understand the following work in order to do it. 4. MINOS AND COFACTOS If we cross out one row and one column of a matrix and find the determinant of the remaining array, we have the minor. The minor is designated M and the subscript is the number of the row and column eliminated. The cofactor is numerically the same as the minor but changes sign for every position in the row or column and the change in sign is indicated by the pattern shown. This is designated with a letter corresponding to the elements so in this case the minor would be A and from the sign pattern we find A = M. Consider how we find the determinant of the following x matrix. STEP Put a line through row and column leaving the elements shown. Find the determinant of the x array enclosed in the square. This is called the MINO of a and designated M. The COFACTO is A = M STEP Put a line through row and column leaving the elements shown. Form these into a x array and find the determinant. The result is the Minor M. The COFACTO is A = - M STEP Put a line through row and column leaving the elements shown. Form these into a x array and find the determinant. The result is the Minor M. The COFACTO is A = M The determinant of the whole array is now found from : A = a A + a A + a A For larger arrays the method is the same but the process is repeated until we are left with a x array. The cofactors take on the sign as indicated by the element position shown. D.J. Dunn

4 WOKED EXAMPLE No. Find the determinant of A a = A = - a = - A = - a = 4 A = - A = a A + a A + a A = () (-) + (-)(-) + (4)(-) = - SELF ASSESSMENT EXECISE No. Find the determinant of the following x matrices. Answers A = 4 B = -8 C = - D.J. Dunn 4

5 . ADJOINT MATIX Another concept used in matrix methods is the Adjoint or Adjugate matrix. This has very useful properties in the solution of problems. This is a matrix formed from all the cofactors of the original matrix and then transposed. We designate this with adj If we had x matrix designated A, the Adjoint is given as: WOKED EXAMPLE No.4 Find the adjoint of the x matrix shown. Go on to find the product of A and adj A First find all the cofactors. emember to use the pattern for the sign changes. Now make a new matrix and transpose it to find the Adjoint. Now multiply A by adj A D.J. Dunn

6 An important point to emerge here is that A adj A always produces a square matrix with all elements zero except the leading diagonal which has all the same elements. ecall that: If a matrix is multiplied by a constant, all the elements are multiplied by a constant. A unit matrix is one with all elements zero except the leading diagonal in which the elements are. The determinant of the unit matrix is. It follows that the product of a constant (A) with the unit matrix is as shown. The determinant of a matrix with all the leading elements A and all other elements zero, is A. Hence if a unit matrix is multiplied by det A, the elements of the leading diagonal will all be det A It also follows that A adj A = det A x the unit matrix. i.e. In the last example, the determinant of matrix A is clearly - and this could be verified by finding det A by the usual method. 6. INVESE MATIX We are on the last stage now and next we will be able to solve simultaneous equations. Suppose we had two matrices A and B such that the product is the unit matrix, i.e. AB = I and it follows that BA = I Matrix B is the inverse of matrix A so we denote it A and replace B with this, so AA = I We have already used the relationship A adj A = A I So equating we have The important result is A adj A = A AA adj A = A A A = (adj A)/ A D.J. Dunn 6

7 WOKED EXAMPLE No. Find the inverse of A given First find all the cofactors. Next find the adjoint of A Next find the determinant of A A =()() + ()() + ()() = 4 Now find the inverse of A A = (adj A)/ A D.J. Dunn

8 . SOLVING SIMULTANEOUS EQUATIONS You will find more useful examples on this at Consider the three equations x y + z = x y + z = 4 -x + y - z = - A solution exists if we can manipulate all the coefficients to the form x + y + z = a x + y + z = b x + y + z = c The matrix form would be as shown. It would follow that x = a, y = b and z = c AUGMENTED MATIX If we write the matrix in this form it is called an augmented matrix. This makes it easier for us to manipulate the figures and turn the coefficient matrix into a unit matrix. emember that we can do the following to the simultaneous equations without changing the validity. ) We can swap the order of the equations as written down without materially changing anything. ) We can multiply any equation by a constant and the equality is maintained. ) We can form new equations by adding a multiple of any one to another. In the augmented matrix this means we can perform the same operations on the rows (not forgetting that it apples to the fourth column as well). The symbol means 'swap' and means 'becomes'. WOKED EXAMPLE No.6 Solve x - y = 4 and x + y = Write the augmented matrix This method requires that we have a as the first element. This can be done in this case by swapping the rows. Next we need to make the element below the into a zero. This is achieved by multiplying row by - and adding it to row to form a new row. This is within the rules explained above. Now we must make the last element into by dividing the second row by - Now the last operation is to make the into a and this can be done by multiplying row by - and adding it to row. The solution is x = 4, y = - D.J. Dunn 8

9 WOKED EXAMPLE No. Solve x + y - z = x - y + z = x y - z = Write the augmented matrix and manipulate / / / / / + + hence x =, y = - and z = This method requires a bit of intuition so a method that uses strict rules is covered next. D.J. Dunn 9

10 SOLVING WITH INVESE MATICES We started the tutorial by saying if a matrix B existed such that AB = I then the numerical solution for X is X = B b The matrix B is the inverse of A so we have X = A b The next worked example uses the same material as the last worked example. WOKED EXAMPLE No.8 Solve the x, y and z given the three simultaneous equations. x + y + z = 8 x y z = -4 x + y z = 6 Create the following matrix. Find the inverse of A using the method outlined earlier. Now solve X = A b emember the multiplication rule and note we divide the result by 4 ow x column = ¼{()(8) + ()(-4) + ()(6)} = ¼{6-8} = ow x column = ¼{()(8) + (-)(-4) + ()(6)} = ¼{} = ow x column = ¼{()(8) + ()(-4) + (-)(6)} = ¼{} = The solution is x =, y = and z = These values may be substituted back into the original equations to check them out. D.J. Dunn

11 SELF ASSESSMENT EXECISE No. Solve the following simultaneous equations. x + y + z = x y z = x + y z = -. x - y - z = 6 x + y = x + y z = 6. x + y + z = x + y z = - x + 4y z = x y = -9 x + y =. x y = 8 x + y = 4 Answers are on the next page. D.J. Dunn

12 ANSWES to SAE. x =, y = - z =. x = y = - z =. x = y = - z = 4. x =. y = 4. x = 4 y = -4 D.J. Dunn