c1,1c1,2

adj( A) = .

.c1,n

c	2,1	.	.	c	n	,1
c2,2		.	.	cn,2
	.				.
	.				.
c2,n		.	.	cn,n

Calculating the Inverse of a Matrix

After the previous slightly complex definitions, the calculation of the inverse matrix is relatively simple.

A−1 = adjA( A)

Clearly, if the determinant of A is zero, the inverse cannot be calculated and the matrix is said to be singular.

5 Application – Solving Linear Equations

One application of matrices is in solving linear equations (or simultaneous equations as they are often known). For example:

x + y + z = 6

2x +3y +4z = 20 4x +2 y +3z =17

This can be written in terms of matrices:

	1	1	1 x			6
	2	3	4				20
				y		=
	4	2	3				17
				z

or more generally

A×X = R

To solve this we simply need to pre-multiply both sides by the inverse of A

A−1 × A× X = A−1 ×R

X = A−1 ×R

In this case the answer is

13				− 13		13			6			1
	3 13			− 13					20			2
						2 3					=
	2	2	3	2	3	1	3		17			3

Eigenvalues (http://www.euclideanspace.com/maths/algebra/matrix/functions/eigenv/in dex.htm)

The eigenvalues of a matrix [M] are the values of such that:

[M] {v} = {v}

where {v} = a vector

this gives:

|M - I| = 0

where I = identity matrix

this gives:

so

(m11- ) (m22- ) (m33- ) + m12 m23 m31 + m13 m21 m32 - (m11- ) m23 m32 - m12 m 21 (m33- ) - m13 (m22- ) m31 = 0

the values of are the eigenvalues of the matrix.