Unformatted text preview: duct of A’s row 1 and B’s
column c31 = a31b11 +a32b21
14 Powers of Matrices
If A is a square matrix, it is possible to write
A3=AAA, et cetera.
n times 1 − 1
Example: A = Example: 0 1 1 −2
A = AA = 0 1 2 1
n 1 − 3
A = A A= 0 1 3 2 −n 1 15 Multiplication of Column by Row Vector
Re-cap: the inner product of the two vectors can be thought of as a
multiplication of a row vector by a column vector:
multiplication v1 u ' v = [ u1u 2u3 ] × v2 = u1 × v1 + u2 × v2 + u3 × v3 v3 What happens if we multiply a column vector by a row vector?
u1 uv ' = 2 ×[v1v2 v3 ] =
? u3 Remembering the multiplication conformability requirement, the result of
this multiplication will be a matrix dimension 3x3:
uv' = u 2 v1 u3v1 u1v2
u 2 v2
u 3 v2 u1v3 u 2 v3 u3v3 16 Matrices and Systems of Linear Equations
Matrix notation makes describing systems of linear equations very easy!
A = 1 4 3
−1 1 x1 − 2, x = x2 x3 5 6 x1 + 3 x2 + x3 Ax = x1 + 4 x2 − 2 x3 4 x1 − x2 + 5 x3 Ax is thus a column vector (dimension 3 x
1) where each element is the left hand
side of one of the three equations.
side To complete the description of the system of three linear equations we
need to add the system’s right hand side:
The original system is: 6 x1 +3 x2 +x3 =22 x1 +4 x2 −2 x3 =12 x −x +5 x =10
1 6 x1 + 3 x2 + x3 22 x + 4 x − 2 x = d = 12 ,
Ax = 1 2
3 10 4 x1 − x2 + 5 x3 Ax = d Matrix A is often referred to as
coefficient matrix. 17 National Income Model
Consider a national income model in two endogenous variables Y and C:
Consider Y =C +I 0 +G0 C =a +bY Y
We re-write it in the form −C = I 0 + G0 −bY + C = a The corresponding coefficient matrix for this system will be −1
− b I + G The right hand side column vector
d = 0 0 Ax = d a
18 Division of Matrices (x)
Just like we do it with numbers, we can add, subtract and
multiply matrices (subject to conformability conditions).
multiply However, we cannot divide matrices!
divide One of the reasons why the operation of division is not defined
for matrices is that, as we shall see later, in general AB ≠ BA 19 The
i =1 i Σ Σ Notation (read: sigma) is used as a shorthand for summation.
6 ∑y = x1 + x2 + x3 j=3 ∑1 axi = ax1 + ax2 + ax3 = a ∑1 xi i= i= 3 3 3 ∑a
j =1 j j = y3 + y 4 + y5 + y 6 x j = a1 x1 + a2 x2 + a3 x3 Matrix multiplication shorthand: C =A B m× n×
n cij =
1 20 Commutative, Associative and Distributive Laws
Commutative law: a+b=b+a (addition) ab=ba (multiplication)
Associative law: (a+b)+c=a+(b+c) Distributive law: (ab)c=a(bc) a(b+c)=ab+ac MATRICES
Distributive Multiplication A+ B = B + A
AB ≠ BA
( A + B ) + C = A + ( B + C ) ( AB )C = A( BC )
A( B + C ) = AB + AC
( B + C ) A = BA + CA 21 AB ≠BA
Consider multiplying these two matrices: 1
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