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- DETERMINANTS, INVERSE OF A MATRIX
- Reference Croft Davision, Chapter 7, Blocks

3,4 - http//www.math.utep.edu/sos/math
- Determinant
- All square matrices, A, possess a determinant

denoted by - det(A), A.
- Determinant of a 2x2 matrix

If , then

det(A) A ad - bc

A matrix which has a zero determinant is called

singular. Minors and cofactors of a 3x3

matrix Let aij be an element of a matrix A. The

minor of aij is the determinant formed by

crossing out the ith row and jth column of

det(A). The cofactor of aij (-1)ij x (minor

of aij) Note that the term (-1)ij is called the

place sign of the element on the ith row and jth

column. The following may help you to memorize

this.

Determinant of a 3x3 matrix Consider a general

3x3 matrix, A Det(A) can be calculated by

expanding along any row or column. For example,

expanding along the first row

A a11x(its cofactor) a12x(its cofactor)

a13x(its cofactor)

e.g.1 Find the value of and

- Properties of determinants
- i. If every element of a given row (or column)

of the square matrix is multiplied by the same

factor, the value of the determinant is

multiplied by that factor - ii. If B is obtained by interchanged any 2

rows (or columns) of A, then - B -A
- Adding or subtracting a multiple of one row (or

column) to another row (or column) leaves the

determinant unchanged. - iv. If A and B are 2 square matrices and that AB

exists, then det(AB) det(A)det(B).

v. If 2 rows or 2 columns of a square matrix are

equal, the determinant of the matrix is

zero. Exercise p.390 Q1a, 4, 5

Inverse of a Matrix The inverse matrix of a

square matrix A, usually denoted by A-1, has the

property AA-1 A-1A I Note that if

A 0, A does not have an inverse. A ? 0, A

does have an inverse

Finding the inverse of a matrix The followings

are steps to find the inverse of a matrix A when

A ? 0, i. Find the transpose of A, denoted

AT. ii. Replace each element of AT by its

cofactor. The resulting matrix is called the

adjoint of A, denoted adj(A). iii. e.g. 2

Find the inverse of

Application of matrices in computer graphic

(Brief Introduction) Matrices can be used to

represent points, lines and even figures

(graphics). In a 2 dimensional space

represents a point with coordinates

(x0, y0),

represents a straight line with end points (x0,

y0) and (x1, y1)

Similarly can be used to

represent the following figure.

Now, you have seen how graphics can be stored

as matrices. Once a graphic is represented by a

matrix, they can be easily inputted into computer

for storage and manipulation. Manipulation

(include scaling, rotation, reflectionetc.) on

the figure can now be treated as a mathematical

process on the matrices (transformation) which

can be done (easily and fast) by a

computer. However, the details of transforming a

graphical matrix is out of our scope and will not

be discussed here.

Exercise p.398 Q2, 3e,10a, 11a

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