A national income model in two endogenous variables y

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Unformatted text preview: B = 6 4 −1 7 AB ≠ BA Calculate first the product AB: 1× 0 + 2 × 6 1× − 1 + 2 × 7 12 13 AB = = 24 25 3 × 0 + 4 × 6 3 × − 1 + 4 × 7 Now reverse the order of operands: 0 ×1 −1×3 BA = 6 ×1 + 7 ×3 0 × 2 −1× 4 − 3 = 27 6 × 2 + 7 × 4 − 4 40 22 Identity Matrices Definition: Identity matrix is a square matrix with 1s in its principal Identity square diagonal and 0s everywhere else. diagonal Notation: I n or I where n is the matrix’ dimension. 1 I = I 3 = 0 0 Properties: 0 1 0 IA =AI =A I k =I 0 0 1 Watch out for the dimension of I! 1 IA = 0 01 2 3 1 2 3 = =A 1 2 0 3 2 0 3 1 0 0 1 2 3 1 0 3 AI = 0 1 0 = 2 0 3 = A 2 0 3 0 0 1 23 Null Matrices Definition: A null matrix (zero matrix) is a matrix of arbitrary dimension of all of whose elements are zeroes. all Notation: 0 Properties: 0 0= 0 = 2x2 0 0 0 0 0 0 0 = 0 = 0 0 0 3x3 0 0 0 A+0 = 0+ A = A A× 0 = 0× A = 0 24 Idempotent Matrices Definition: A matrix is idempotent if its product with itself remains itself. AA = A Obviously, an identity matrix is idempotent: Obviously, An example of an An idempotent matrix: idempotent 1 0 0 0 1/ 2 1/ 2 II = I 0 1 / 2 1 / 2 25 Idiosyncrasies of Matrix Algebra Identity matrices are similar to the number 1 in scalar algebra: Identity multiplication by an identity matrix does not change the matrix (however, watch out for dimensions!) watch Null matrices are similar to zeroes in scalar algebra. Again, dimensions Null do matter. do Very importantly, the commutative law for multiplication does not Very commutative hold in case of the matrix algebra: BA AB ≠ Also, the product of two nonzero matrices is not necessarily zero (as Also, two A = 0 it would be the case with scalars): it 2 4 − 2 4 AB = 0 ⇒ / AB = =0 B = 0 1 2 1 − 2 Another idiosyncrasy: CD = CE ⇒D = E / 2 C = 6 3 1 D = 9 1 CD = CE , D ≠ E 1 −2 E = 3 2 1 2 26 Transposes Remember the distinction between a row vector and a column vector: Column vector: x1 x= x 2 Row vector: x' = ( x1 x2 ) We can say that one can be obtained from another by interchanging We columns with rows. columns () Definition: Matrix A' AT is called a transpose of matrix A if it is ' transpose obtained from the latter by interchanging columns and rows, i.e. if aij 3 8 9 A= 1 0 4 3 1 A' = 8 0 9 4 = a ji Transposes have a “reverse” dimension compared to the original matrix: if A Transposes is dimension 2x3, its transpose is dimension 3x2. Square matrices do not change their dimension when transposes are taken. change 27 Symmetric Matrices Definition: A square matrix that is equal to its transpose is called a symmetric matrix. symmetric An example of a symmetric matrix: 1 0 4 D = 0 3 7 4 7 2 It is easy to observe that in matrix D all elements are the symmetric It with respect to the principal diagonal: d ij = d ji with 28 Properties of Transposes The obvious property: The easy property: ( A') ' = A ( A + B )′ = A'+B' ( AB )′ = B′A′ The dimensionality argument: suppose A , B The The reverse order prope...
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