Basic matrix operation

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• Basic matrix operation
• Reference Croft Davision, Chapter 7, Blocks
  1,2
• http//www.math.utep.edu/sos/math
• Introduction to Matrices

A matrix is a rectangular array or block of
numbers usually enclosed in brackets A m x n
matrix has m rows and n columns If the matrix A
has m rows and n columns we can write
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where aij represents the number or element in the
ith row and jth column. Special Matrices A
square matrix has the same number of rows as
columns. The main diagonal of a square matrix is
the diagonal running from top left to bottom
right. An identity matrix, denoted by I, is a
square matrix with ones on the main diagonal and
zeros elsewhere.
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The transpose of A is obtained by writing rows as
columns and columns as rows, and is denoted AT
Equality of Matrices If A (aij) and B (bij),
A B if and only if aij bij
Addition and Subtraction of Matrices Matrices of
the same size may be added to and subtracted from
one another. To do this, the corresponding
elements are added or subtracted.
e.g. 1 e.g. 1 If ,
find A B, BC and
B-C .
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Multiplication of a Matrix by a number Any matrix
can be multiplied by a number. To do this, each
element of the matrix is multiplied by the
number. e.g.2 If , find 2A,
-A.
Multiplication of Matrices If A is a nxm matrix
and B is a pxq matrix. For the product AB to
exist we must have mp, i.e.
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Note that matrix multiplication is i. not
commutative (i.e. AB?BA). ii. associative i.e.
ABC(AB)CA(BC) . iii. If CAB, the element cij
is found from row i of A and column j of B, as
follows
e.g. 3 If
, find AB
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Note that when a square matrix is post- or
pre-multiplied by an identity matrix of the
appropriate size the matrix is unchanged,
i.e. AI IA A
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Exercise p. 364 Q1, 3, 4, 12, 15 p. 373 Q7,
10, 13