|
| Basic
matrix
operations |
| Matrix addition and
subtraction |
| Matrices
can be added or subtracted only if they have the same
dimensions. |
| Addition |
| The
elements of the sum of the two matrices with the same
dimensions, A
and B,
are equal to the sums of |
| the
corresponding elements aij
+ bij
that is, |
| A
+ B
= [aij
+ bij]. |
| Subtraction |
| The
elements of the difference of the two matrices with the same
dimensions, A
and B,
are equal to the |
| differences
of the corresponding elements aij
-
bij
that is, |
| A
-
B
= [aij
-
bij]. |
| Example:
Given are matrices, A
and B, find the
sum A
+ B
and the difference A
-
B.
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and |
 |
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then, |
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and |
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| Scalar
multiplication of a matrix |
| Multiplication
of a matrix A
by a scalar c
is defined as |
| c
· A = [c · aij], |
| that
is, each element of the matrix is multiplied by c.
Therefore, if c
= 0, the result is the null
matrix. |
| Example:
Given the matrix |
 |
find the
product -
2 · A. |
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| Matrix
multiplication |
| Multiplication
of two matrices is defined only if the number of columns of the
first matrix equals the number of |
| rows
of the second. Matrices that satisfy this condition are said to
be conformable. |
| If
A
= [aij]
is a matrix of order m
´
n
and B
= [bjk]
is a matrix of order n
´
p,
then the product |
| C
= AB |
| where
C
= [cik]
is the m
´
p
matrix of which the ikth
entries are defined by the formula |
 |
| Therefore, the
ikth
entry of the product equals the inner product of the components
of the ith
row of the first |
| matrix
with the components of the kth
column of the
second matrix. |
| Example:
Given are conformable matrices, A
and B, find the
product AB.
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and |
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| then, |
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| or |
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| The
matrix-vector product |
| If
A = [aij]
is an m
´
n
matrix, x =
{xj}
a column vector of order n,
and y = {yi}
a column vector of order m |
| then,
y = Ax
is the matrix-vector product, |
| where |
 |
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| The
column dimension of the matrix A
must equal the row dimension of the vector x
to obey the product |
| definition. |
| Example:
Given is the matrix A
of order 2
´
3
and the column vector x
of order 3, find the
product Ax.
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and |
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| then, |
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| If A is
a square matrix of order n
then,
|
| AI =
IA = A |
| where
I is the
identity
matrix of order n. |
|
| The
determinant of a matrix |
| Determinants
are defined only for square matrices. |
| The
determinant D
of an n
´
n
matrix A,
also denoted det(A)
or | A |
is an ordered square array of elements |
 |
| the
value of which is given by an alternating sum of products of the
elements of A,
that can be obtained |
| by
using the method of expanding the determinant to
cofactors. |
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| Pre-calculus contents
C |
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