2019-Midterm-Resit-Solutions

Page 2 - Matrix M and Inverse Calculations

1. Given Matrix M

• Matrix M:[ M = \begin{bmatrix} 0 & 1 & 0 \\ a & b & c \\ 1 & 0 & 1 \end{bmatrix} ]

• Parameters: a, b, c

(a) Compute Determinant (|M|)

• Determinant of M:[ |M| = c - a ]

(b) Inverse Existence Condition

• M^(-1) exists if:

  • Condition: [ c
    eq a ]

(c) Compute Matrix of Cofactors of M

• Cofactors:[ \begin{bmatrix} b & c - a & -b \\ -1 & 0 & 1 \\ c & 0 & -a \end{bmatrix} ]

(d) Compute Inverse of Matrix M

• Inverse M:[ M^{-1} = \begin{bmatrix} -b & \frac{a-c}{1} & 1 \\ \frac{a-c}{1} & -c & 0 \\ 0 & 0 & \frac{b}{a-c} & -1 & a \end{bmatrix} ]

(e) Solve the System ( MV = B )

• Given:

  • Vector V: [ V = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} ]

• Solution:[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = M^{-1} B = \begin{bmatrix} 1 \frac{a-c}{1} \ -\frac{1}{a-c} \end{bmatrix} ]

Page 3 - Macroeconomic Model

2. National Output Model

• Model:[ Y = C + I + G0 ]

• Consumer Expenditure:[ C = bY - T0 ]

• Investment:[ I = 0.4Y + 1 ]

• Endogenous Variables: Y (national income), C (consumption), I (investment)

• Parameters: T0 (tax burden), G0 (government spending), b.

(a) Matrix Formulation

• Matrix Equation:[ \begin{bmatrix} 1 & -1 & -1 \\ b & -1 & 0 \\ 0.4 & 0 & -1 \end{bmatrix} \begin{bmatrix} Y \\ C \\ I \end{bmatrix} = \begin{bmatrix} G0 \\ T0 \\ -1 \end{bmatrix} ]

(b) Compute Determinant (|A|) and Rank of A

• Determinant:[ DET(A) = 0.6 - b ]

• Rank(A) Conditions:

  • Rank 3 if ( b
    eq 0.6 )

  • Rank 2 if ( b = 0.6 )

(c) Unique Solution Condition

• Unique solution when:[ b
  eq 0.6 ]

(d) Cramer’s Rule for Equilibrium Consumption (C∗)

• Consumption Equation:[ C = \frac{DET \begin{bmatrix} 1 & G0 & -1 \\ b & T0 & 0 \\ 0.4 & -1 & -1 \end{bmatrix}}{0.6 - b} ]

(e) Conditions for Infinite or No Solution

• Rank Analysis:

  • For infinite solutions:

    • Condition: [ -0.6G0 - 1.4T0 - 0.6 = 0 ]

  • For no solutions:

    • Condition: [ -0.6G0 - 1.4T0 - 0.6
      eq 0 ]

Page 4 - Matrix H and Vectors

3. Matrix H Characteristics

• Matrix Definition:[ H = \begin{bmatrix} 1 & 0 & a \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & b \end{bmatrix} ]

(a) Rank Condition for ( Rank(H)=4 )

• Conclusion:

  • No, only 3 columns available.

(b) Rank Conditions for Values of a and b

• Rank(H) = 3:

  • Condition: [ a
    eq 0 \text{ or } b
    eq 1 ]

• Rank(H) = 2:

  • Condition: [ a = 0 ext{ and } b = 1 ]

(c) Linear Combinations and Basis Formation

• Given Vectors:[ V1, V2, V3, V4 ]

• Theorem:

  • For values of b:

    • Condition for base in R4: [ b
      eq 1 ]

• Determinant for the basis:[ det(V1, V2, V3, V4) = 1 - b ]