DETERMINANTS, INVERSE OF A MATRIX

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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
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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.
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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)
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e.g.1 Find the value of and
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• 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).

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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
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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
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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
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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.
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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