Shopping and bought 2 loaves of bread at 5 each and 3

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Unformatted text preview: duct of A’s row 1 and B’s column 1. column c31 = a31b11 +a32b21 14 Powers of Matrices If A is a square matrix, it is possible to write If A2=AA, =AA, A3=AAA, et cetera. =AAA, In general: In An=AAAAAAAA…A n times 1 − 1 Example: A = Example: 0 1 1 −2 A = AA = 0 1 2 1 A = 0 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 Re-cap: 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 ] = u ? u3 Remembering the multiplication conformability requirement, the result of Remembering this multiplication will be a matrix dimension 3x3: this u1v1 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! 6 A = 1 4 3 4 −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 Ax 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 To need to add the system’s right hand side: need The original system is: 6 x1 +3 x2 +x3 =22 x1 +4 x2 −2 x3 =12 x −x +5 x =10 4 2 3 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 1 1 A = − b I + G The right hand side column vector The d = 0 0 Ax = d a 18 Division of Matrices (x) Just like we do it with numbers, we can add, subtract and Just multiply matrices (subject to conformability conditions). multiply However, we cannot divide matrices! However, divide One of the reasons why the operation of division is not defined One for matrices is that, as we shall see later, in general AB ≠ BA 19 The Symbol Symbol 3 ∑x 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 l n cij = a ∑ik bkj k= 1 20 Commutative, Associative and Distributive Laws REAL NUMBERS REAL 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 Addition Commutative Associative 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 A = 3 2 0 ,...
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