Points about matrices
Clement Radcliffe,Contributor

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 ? 2 matrices. You just need to continue practising the principle: rows multiply by columns .
• Squaring the 2 ? 2 matrix A is found by multiplying A ? A.
• The determinant of a matrix has value ad- bc.
• The value of determinant of a singular matrix is zero.

All of the above is illustrated by the solutions to the homework given last week.

Homework

1. Matrix C (6/5 2/p) is a singular matrix. Calculate the value of P.

Solution

As C is a singular, then the value of the determinant of C is zero

Give the determinant (a/c b/d) then its value ad - bc = 0

:. 6 x P - 2 x 5 = 0

:. 6P - 10 = 0

:. 6P = 10 or P = 5/3

2. The matrix H = (h/2 2/-h)

(i) Determine H2

(ii) Evaluate h, if H2 = (1/0 0/1)

Solution

1. Given that H = (h/2 2/-h)

Be reminded that the product of H₂ x 2 x H₂ x 2 is a 2 x 2 matrix

H2 = (h/2 2/-h) x (h/2 2/-h)

= ( (hxh + 2x2)/(2xh + -hx2) (hx2 + 2x-h)/(2x2 + -hx-h) ) = ( (h² + 4)/0 0/4+ h²) )

(ii) Since H² = 5 (1/0 0/1)

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

Let us now proceed to use matrices to solve simultaneous equations. The concepts involved are as follows:

Simultaneous equation

• Simultaneous equations are expressed in matrix form AX = B where A is the 2 x 2 coefficient matrix, x is the 2 x1 matrix (x y) and B the 2 x 1 matrix of the constant terms.
• The 2 x 2 coefficient matrix A is converted to the unit matrix by pre-multiplying both sides by the Inverse of A.
  :. A-1 ? A ? X = A-1 B.
• By simplifying both sides, the equation of two 2 ? 1 matrices remains.
• 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 C « X = D

(b) Given the 2 ? 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

-3 ? 4 -2 ? 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 by the inverse

:. 1 4 -2 -3 2 x = 1 4 -2 ? -11

-22 -5 4 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.

1. 2x + 5y = 6

3x + 4y = 8

2. 3x + 4y = 10

4x + 2y = 10

Clement Radcliffe is the principal of Glenmuir High School in May Pen.