To solve for the equilibrium levels of $Y$ (national income) and $C$ (consumption) using matrix algebra, we follow these steps:

Step 1: Express the equilibrium equation

The equilibrium condition is given as: $Y = C + I$ Substitute $C$ and $I$ into the equation: $Y = (100 + 0.8Y) + 50$ Simplify: $Y = 150 + 0.8Y$

Step 2: Rearrange into standard linear form

Rearrange to isolate $Y$ on one side: $Y - 0.8Y = 150$ $0.2Y = 150$

Step 3: Solve for $Y$

Divide through by $0.2$: $Y = \frac{150}{0.2} = 750$

Step 4: Solve for $C$

Now that $Y = 750$, substitute it into the consumption function $C = 100 + 0.8Y$: $C = 100 + 0.8(750)$ $C = 100 + 600 = 700$

Final Results

The equilibrium levels are: $Y = 750, \quad C = 700$

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Expanding Questions:

1. How does the marginal propensity to consume ($0.8$) affect equilibrium income?
2. What happens to $Y$ and $C$ if investment ($I$) increases to $100$?
3. Can this problem be solved graphically, and what would the graph look like?
4. How do changes in the autonomous consumption ($100$) affect the results?
5. What role does the multiplier effect play in determining $Y$?

Tip: The multiplier is calculated as $\frac{1}{1 - MPC}$, which can provide insights into how changes in investment or autonomous spending affect national income.