GATE CS 2022 Q35 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To solve the system using $\text{LU}$ decomposition, we first write the system in matrix form $Ax = b$:

Using Doolittle's algorithm, we set the diagonal elements of $L$ to $1$ ($L_{11} = L_{22} = L_{33} = 1$). Thus, we have $A = LU$:

By multiplying the matrices and equating the elements:

• First row: $U_{11} = 1$, $U_{12} = 1$, $U_{13} = -2$
• Second row:
  • $L_{21}U_{11} = 1 \implies L_{21} = 1$
  • $L_{21}U_{12} + U_{22} = 3 \implies 1(1) + U_{22} = 3 \implies U_{22} = 2$
  • $L_{21}U_{13} + U_{23} = -1 \implies 1(-2) + U_{23} = -1 \implies U_{23} = 1$
• Third row:
  • $L_{31}U_{11} = 2 \implies L_{31} = 2$
  • $L_{31}U_{12} + L_{32}U_{22} = 1 \implies 2(1) + L_{32}(2) = 1 \implies L_{32} = -\frac{1}{2}$
  • $L_{31}U_{13} + L_{32}U_{23} + U_{33} = -5 \implies 2(-2) + \left(-\frac{1}{2}\right)(1) + U_{33} = -5 \implies U_{33} = -\frac{1}{2}$

Now, we solve the system of equations to find $x_1$:

1. $x_1 + x_2 - 2x_3 = 4$
2. $x_1 + 3x_2 - x_3 = 7$
3. $2x_1 + x_2 - 5x_3 = 7$

Subtracting equation (1) from (2) gives: $$2x_2 + x_3 = 3 \quad \text{--- (4)}$$ Multiplying equation (1) by $2$ and subtracting from (3) gives: $$-x_2 - x_3 = -1 \implies x_2 + x_3 = 1 \quad \text{--- (5)}$$ Solving equations (4) and (5) yields $x_2 = 2$ and $x_3 = -1$. Substituting these back into equation (1): $$x_1 + 2 - 2(-1) = 4 \implies x_1 = 0$$

Thus, $L_{32} = -\frac{1}{2}$, $U_{33} = -\frac{1}{2}$, and $x_1 = 0$.