Lecture 3
LINEAR ALGEBRA

• Matrices
• Determinants
• Linear System of Equations
• Cramer's Rule

Matrices

• A matrix (call it A) is a rectangular array of numbers, a_ij
  A = [a_ij],
  i = 1, 2,..., m
  j = 1, 2,..., n
  

• Dimension of a matrix
  • m x n matrix: m rows and n columns

• Matrix operations
  • Addition and subtraction of two matrices A and B
  • Matrices must have the same dimension
  • a_ij + b_ij = c_ij

• Scalar multiplication (kA)
  • Each element of the matrix is multiplied by the given scalar
  • kA = [ka_ij]

• Multiplication of matrices (AB)
  • Check: Is Column dimension of A = Row dimension of B?
  • A_mxn B_nxk = C_mxk
  • Matrix multiplication is not commutative: AB does not equal BA

• Identity matrix (I)
  • Square matrix with 1's in its principal diagonal and 0's elsewhere
  • A_mxn I_nxn = A_mxn

• Transpose of a matrix (A^T)
  • The rows and columns of A_mxn are interchanged
  • Dimensions of A^T: nxm
  
  

   

Determinants

• The determinant of a square matrix A
• Notation: det (A)
• Uniquely defined scalar associated with matrix A
• Second-order determinant
  • Consider the A_2X2 matrix:

    a11	a12
    a21	a22

  The determinant of matrix A is:
  • det (A) = a₁₁a₂₂ - a₂₁a₁₂

  Note: If the rows of the matrix are linearly dependent, the determinant is zero.

• Third-order determinant
  Consider the A_3x3 matrix:

  a11	a12	a13
  a21	a22	a23
  a31	a32	a33

  Obtain the determinant.

• Minor of an element, M_i
  • The determinant obtained by deleting the i-th row and j-th column

• Cofactor of an element, C_ij
  • Same magnitude as the minor of the element
  • Sign given by i+j:
    • If i+j even, same sign as minor
    • If i+j odd, cofactor has opposite sign

• Third-order determinant revisited
  • det (A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃

• Inverse of a matrix (A^-1)
  • A must be a square matrix
  • AA^-1 = I
  • The inverse exists only if det(A) does not equal 0
  • Singular matrix:
    The inverse does not exist
  Steps
  1. Find the determinant of the matrix.
  2. If det (A)=0, inverse does not exist
  3. Obtain the cofactors of all elements and form the cofactor matrix, [C] = [/Cij/].
  4. Take the transpose of [C] to get adjoint of A, adj A.
  5. Divide adj A by det(A) to obtain the inverse of A.

  Example 1. Consider the matrix A:

  1	2
  3	4

  1. Obtain the determinant of matrix A:
     det (A) = -2

  2. Obtain the cofactor matrix.
     • Obtain the cofactor of each element: C₁₁ = 4    C₁₂ = -3    C₂₁ = -2    C₂₂ = 1
     • Form the cofactor matrix.

  3. Obtain the adjoint of A by taking the transpose of [C].

  4. Obtain the inverse of A.

  5. Verify that A A^-1 = I.




Linear systems of equations

• Consider the following system of equations:
  • A_nxn X_nx1 = B_nx1       (1)
  • The matrix A is called the coefficient matrix.
  • The solution vector is obtained by premultiplying each side of equation (1) by A^-1:
    X = A^-1 B
  • Note: For this to work, the inverse of A must exist, i.e. det(A) should not equal 0.

• Cramer's rule
  How do you obtain the solution for the j-th variable, x_j in equation (1)?

  Steps

  1. Replace the j-th column in the coefficient matrix by the vector B.
  2. Compute the determinant of the altered matrix, det (A_j).
  3. Compute the determinant of A, det(A).
  4. x_j = det (A_j) / det (A).

• Example 2
  Solve the following system of equations:
  -x + 3y + 2z = 24
  x + z = 6
  5y - z = 8

  1. Obtain the determinant of the coefficient matrix, det (A):
     det (A) = 18

  2. To obtain x:
     1. Replace Column 1 by the vector B.

     2. Compute the determinant, det (A₁):
        det (A₁) = -18.

     3. Obtain the ratio det (A₁) / det (A):
        x = -1.

  3. Find y and z.
     • det (A₂) = 54
     • det (A₃) = 126
     • y = det (A₂) / det (A) = 3
     • z = det (A₃) / det (A) = 7

  4. Verify that the values of x, y and z satisfy the equations.

• Example 3
  Consider the following system of equations:
  x + 3y = 7
  2x - y = 0
  
  1. Write the equations in matrix form: AX=B.

  2. Obtain A^-1 and solve for x and y.
     • det (A) = -7

     • det (A₁) = -7

     • det (A₂) = -14

     • x = 1

     • y = 2

  3. Verify that the solutions satisfy the equations.




Application: The Keynesian Model

Equations:
• C = 100 + 0.8(Y-T)
• I = 500 - 50r
• M_d/P = 0.2Y - 25r + 500
• P = 1
• G = 400
• T = 400
• M = 520
Notation:

Y is GNP, r is the interest rate, C is consumption, I is investment, Md is money demand, P is the price level, G is government expenditure, T is taxes, and M represents money supply.

Variables:
• Endogenous variables: Y, r, C, I, M_d
• Exogenous variables: M, G, T, P

Equilibrium conditions:

• Equilibrium in the goods market: Y = C + I + G       (1)
• Equilibrium in the money market: M = M_d           (2)

Questions:

1. Obtain the IS equation from (1).
2. Obtain the LM equation from (2).
3. Write the system of equations in matrix form, AX=B.
4. Compute the determinant of the coefficient matrix.
5. Obtain the inverse of the coefficient matrix.
6. Using X=A^-1B, solve for the equilibrium values of Y and r.
7. Alternatively, use Cramer's rule to obtain the equilibrium values of Y and r.
8. Obtain the equilibrium values of C, I and M_d. Verify that these values satisfy (1) and (2).
9. What happens to Y and r if:
   • T increases by $100?
   • G increases by $100?
   • M increases by $100?

Here's the Keynesian Model in MAPLE.