Core Pure 6: Matrices

Question 1

Method of how to find the Inverse of a 3x3 Matrix

Answer 1

Step 1:
Find the value of det A

Step 2:
Replace each element A with its minor (sub-determinant)

Step 3:
Use this pattern to change the signs of the elements of the matrix of minors:

(+ - +)
(- + -)
(+ - +)

This then becomes the matrix of co-factors.

Step 4:
Transpose this matrix of co-factors.

Question 2

When Solving 3x3 Simultaneous Equations,

How do the planes intersect if detA ≠ 0

Answer 2

The equations are consistent and have one unique solution.

The planes meet at a point. The system of equations is consistent and has one solution represented by the point. This is the only case when the corresponding matrix is non-singular.

Question 3

When Solving 3x3 Simultaneous Equations,

How do the planes intersect if detA = 0

Answer 3

a)
The equations are consistent, but meet in a line with indefinitely many solutions. This is called a SHEAF.

(If the 2 equations are multiples of each other)

The planes form a sheaf. The system of equations is consistent and has indefinitely many solutions represented by the line of intersection of the three planes.

b)
The equations are inconsistent - i.e they have no solutions. This can happen in 2 ways:

• i)
  They meet in a PRISM
  (If the 2 equations are not multiples of each other)
  The planes form a prism. The system of equations is inconsistent and has no solutions.
• ii)
  Two or more of the planes are parallel
  (i.e multiples of each other)
  Two or more planes are parallel and non-identical. The system of equations is inconsistent and has no solutions.