Matrices – Part 3
AT THE outset, I wish to highlight the following points about matrices. They are vital to your full understanding of this topic. There is no reason to have difficulty in multiplying 2 X 2 matrices. You just need to continue practising the principle: rows multiply by columns both with the same number of elements. Squaring the 2 X 2 matrix A is found by multiplying A X A. The determinant of a 2 X 2 matrix has value ad - bc, where the elements of the matrix are a,b, c, and d.
The value of the determinant of a singular 2 X 2 matrix is zero, that is ad-bc = 0. The above are illustrated by the solutions of the homework given last week.
HOMEWORK
1. Given the matrices A = 3 1 and B= -2 4
2 0 3 -1 Find the values of i) 3A ii) -2B
SOLUTION
i) Given that A= 3 1, then 3A = 9 3 2 0 6 0 ii) Given that B= -2 4, then - 2B =4 -8 3 -1 -6 2 2. Matrix C = 6 2 is a singular matrix. Calculate the value of p. 5 p Given p, find C2 .
SOLUTION
As C is a singular, then the value of the determinant of C is zero. Given the determinant a b, then its value ad - bc = 0 c d Substituting the elements of matrix C, :.6 X P - 2 X 5 = 0 :. 6P - 10 = 0 :. 6P = 10 or P = 5/3 Answer P = 5/3 Since P = 5/3, Then C2 =6 2 6 2 6X 6 + 2 X 5 6X 2 + 2 X 5/3
5 5/3 5 5/3 = 5X 6 + 5/3 X 5 5X 2 + 5/3 X5/3 = 36 + 10 12 + 10/3 46 46/3 30 + 25/3 10 + 25/9 = 115/3 115/9 2. Given the matrix H= h 2 2 -h (i) Determine H2 (ii) Evaluate h, if H2 = 5 1 0
0 1
SOLUTION
(i) Be reminded that the product of H2 X 2 X H2 X 2 is a 2 X 2 matrix H2 = h 2X h 2 2 -h 2 -h = hXh + 2X2 hX2 + 2X-h = h2 + 4 0 2Xh + -hX2 2X2 + -hX-h 0 4+ h2 (ii) Since H2 = 5 1 0 0 1 :. h2 +4 0 = 5 1 0 0 4 + h2 0 1 ( h2 +4) 1 0 = 5 1 0 0 1 0 1 :. h2 + 4 = 5 or h2 = 1 :.h 2 = 1 h= ± 1 Answer h= ± 1 Let us now proceed to use matrices to solve simultaneous linear equations. The method involved is as follows: An important prerequisite is to be able to find the inverse of a 2X2 matrix. Given the matrix A, then the inverse of A is: 1 d -b ad- bc -c a
EXAMPLE
Given that A = 3 6 , find the inverse of A, or A-1 1 5 Using the formula above, A-1 = 1/9 5 -6 -1 3 The simultaneous equations are expressed in matrix form A xX = B, where A is the 2 x 2 coefficient matrix, X is the 2 x 1 matrix x and B the 2 x 1 matrix of the constant terms. y For example, given the equations, 2x + 4y = 7 3x - y = 5 it is expressed as: 2 4 x =7 3 -1 y 5
The 2 ? 2 coefficient matrix A is converted to the unit matrix by pre- multiplying both sides by the Inverse of A :. A-1 XA X X = A-1 B. NB: You must multiply both sides of the equation to maintain the values of the variables.
By simplifying both sides, the equation of two 2 X 1 matrices remain.
Equating terms will enable you to find the values of x and y, the solution of the original simultaneous equations. The above is illustrated by the solution to the following example.
EXAMPLE
Given that -3x + 2y = -11 5x + 4y = 33 (a) Express the simultaneous equations in the form Cx X = D. (b) Given the 2 X 2 matrix C, find: (i) The determinant of C. (ii) The inverse of C.
SOLUTION
(a) -3x + 2y = -11 5x + 4y = 33 is expressed as: -3 2 x -11 5 4 y = 33 I expect that the pattern is clear. (b) (i) As C is -3 2 5 4 , then the determinant is -3X 4 - 2 X 5 = -12 - 10 = -22 (ii) Given the matrix a b, then the inverse is: 1 d -b c d ad - bc -c a :. The inverse of C is: 1/-22 4 -2
-5 -3 The solution of the simultaneous equations is as follows: Given -3x + 2y = -11
5x + 4y = 33 :. -3 2 x = -11 5 4 y 33 Pre-multiply both sides by the inverse of C.
:. 1 4 -2 -3 2 x= 1 4-2 X -11 -22 -5 -3 5 4 y -22 -5 -3 33 1 -22 0 x = 1 -110 -22 0 -22 y -22 -44 :. 1 0 x= 5 0 1 y 2 x= 5 y 2 :.x = 5 and y = 2. Let us now attempt the following example together. (a) Solve the simultaneous equations: 3x + 2y = 1 x + 4y = -3 Expressing the above in matrix form 3 2 x= 1 1 4 y -3 The inverse of 3 2 = 1 4 -2 1 4 10 -1 3 Pre-multiplying both sides of the matrix equation by the inverse of A 1 4 -2 3 2 x= 1 4 -2 1 10 -1 3 1 4 y 10 -1 3 -3 :. 1 0 x= 1 0 1 y-1 x= 1 y -1:. x = 1 and y= -1 Now, please attempt the following for homework. Solve the following simultaneous equations using the matrix method. (a) 4x +y = - 3 (b) 3x - 2y = -1
3x + 2y = -1 4x - y = 2