Sumber:
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/Summary3.html#oper
Basic Definitions
Example
An m n matrix A is a rectangular array of real
numbers with m rows and n columns. (Rows are
horizontal and columns are vertical.) The
numbers m and n are the dimensions of A.
Following is a 4 5 matrix with the
entry A23 highlighted.
0 1 2 0 3
1/3 -1 10 1/3 2
A=
3 1 0 1 -3
2 1 0 0 1
The real numbers in the matrix are called its
entries. The entry in row i and column j is called
aij or Aij.
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Operations with Matrices
Examples
Transpose
The transpose, AT, of a matrix A is the matrix
obtained from A by writing its rows as columns.
If A is an m n matrix and B = AT, then B is the
n m matrix with bij = aji.
Transpose
T
0 1 2
1/3 -1 10
0 1/3
= 1 -1
2 10
Sum, Difference
If A and B have the same dimensions, then their
Sum & Scalar Multiple
sum, A+B, is obtained by adding corresponding
entries. In symbols, (A+B)ij = Aij + Bij. If A and B
have the same dimensions, then their difference,
0 1
1
2
1
1
A - B, is obtained by subtracting corresponding
=
- +2
1/3
entries. In symbols, (A-B)ij = Aij - Bij.
1
2/3
5/3
2
5
Scalar Multiple
If A is a matrix and c is a number (sometimes
called a scalar in this context), then the scalar
multiple, cA, is obtained by multiplying every
entry in A by c. In symbols, (cA)ij = c(Aij).
Product
0 1
1/3
1
1
2/3
2
1
=
2/3 -2
5/3
1/3
Product
If A has dimensions m n and B has dimensions
n p, then the product AB is defined, and has
dimensions m p. The entry (AB)ij is obtained by Visit our Matrix Algebra Tool for
multiplying row i of A by column j of B, which is on-line matrix algebra
done by multiplying corresponding entries
together and then adding the results.
computations.
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Algebra of Matrices
Examples
The n n identity matrix is the matrix I that has
1's down the main diagonal and 0's everywhere
else. In symbols, Iij = 1 if i = j and 0 if i ‚ j.
Following is the 4 4 identity
matrix.
1000
0100
I=
0010
0001
A zero matrix is one whose entries are all 0.
The various matrix operations, addition,
subtraction, scalar multiplication and matrix
multiplication, have the following properties.
A+(B+C) =
(A+B)+C
A+B = B+A
A+O = O+A = A
A+( - A) = O = ( A)+A
c(A+B) = cA+cB
(c+d)A = cA+dA
A=A
0A = O
Additive associative law
Additive commutative
law
Additive identity law
Additive inverse law
Distributive law
Distributive law
Scalar unit
Scalar zero
Multiplicative
A(BC) = (AB)C
associative law
Multiplicative identity
AI = IA = A
law
A(B+C) = AB + AC Distributive law
(A+B)C = AC + BC Distributive law
Multiplication by zero
OA = AO = O
matrix
T
T
T
(A+B) = A + B
Transpose of a sum
Transpose of a scalar
(cA)T = c(AT)
multiple
Transpose of a matrix
(AB)T = BTAT
product
The one rule that is conspicuously absent from
The following illustrates the
failure of the commutative law for
matrix multiplication.
A
=
0 1
1/3
1
1
2/3
2
1
B=
AB =
2/3 -2
-1/3 5/3
BA =
-1/3 2
-2/3 8/3
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this list is commutativity of the matrix product. In
general, matrix multiplication is not
commutative: AB is not equal to BA in general.
Matrix Form of a System of Linear Equations
Example
An important application of matrix multiplication The system
is this: The system of linear equations
x + y - z =4
a11x1 + a12x2 + a13x3 + . . . + a1nxn = b1
3x + y - z = 6
a21x1 + a22x2 + a23x3 + . . . + a2nxn = b2
x + y - 2z = 4
..............
3x + 2y - z = 9
am1x1 + am2x2 + am3x3 + . . . + amnxn = bm
has matrix form
can be rewritten as the matrix equation
1 1 -1
x
AX = B
3 1 -1
y
=
1 1 -2
z
where
3 2 -1
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a11 a12 a13 . . . 1n
a21 a22 a23 . . . a2n
A=
.......
am1 am2 am3 . . . amn
X = [x1, x2, x3, . . . , xn]T
and
B = [b1, b2, x3, . . . , bm]T
Matrix Inverse
4
6 .
4
9
Example
If A is a square matrix, one that has the same
The system of equations
number of rows and columns, it is sometimes
possible to take a matrix equation such as AX = B
124 x
1
and solve for X by "dividing by A." Precisely, a
246 y = 1
square matrix A may have an inverse, written A468 z
1
-1
, with the property that
AA-1 = A-1A = I.
If A has an inverse we say that A is invertible,
otherwise we say that A is singular.
When A is invertible we can solve the equation
AX = B
has solution
x
1
2
4
y =
2
4
6
z
4
6
8
1
1
1
1
by multiplying both sides by A-1, which gives us
X = A-1B.
=
1
-2
-2
2
1
1/2
1
1
1
1 -1/2 0
-2
= 1/2
1/2 .
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Determining Whether a Matrix is Invertible
Examples
In order to determine whether an n n matrix A is
invertible or not, and to find A 1 if it does exist,
write down the n (2n) matrix [A | I] (this is A
with the n n identity matrix set next to it).
The matrix
124
A= 246
468
Row reduce this matrix.
If the reduced form is [I | B] (i.e., has the identity
matrix in the left part), then A is invertible and B
= A-1. If you cannot obtain I in the left part, then
A is singular
is invertible. The matrix
124
B= 246
247
is not.
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Inverse of a 2 2 Matrix
The 2 2 matrix
A=
a b
c d
is invertible if ad - bc is nonzero and is singular
if ad - bc = 0. The number ad - bc is called the
determinant of the matrix. When the matrix is
invertible its inverse is given by the formula
1
d -b
A 1=
ad - bc -c a
.
Example
12
1
1
=
34
=
(1)(4) (2)(3)
.
-2 1
3/2 -1/2
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4
2
1
3
Input-Output Economic Models
An input-output matrix for an economy gives, as its jth column, the amounts (in
dollars or other appropriate currency) of outputs of each sector used as input by sector
j (for one year or other appropriate period of time). It also gives the total production of
each sector of the economy for a year (called the production vector when written as a
column).
The technology matrix is the matrix obtained by dividing each column by the total
production of the corresponding sector. Its ijth entry, the ijth technology coefficient,
gives the input from sector i necessary to produce one unit of output from sector j. A
demand vector is a column vector giving the total demand from outside the economy
for the products of each sector. If A is the technology matrix, X is the production
vector, and D is the demand vector, then
(I - A)X = D,
or
X = (I - A)-1D.
These same equations hold if D is a vector representing change in demand, and X is a
vector representing change in production. The entries in a column of (I - A)-1
represent the change in production in each sector necessary to meet a unit change of
demand in the sector corresponding to that column, taking into account all direct and
indirect effects.
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