Consider a system of linear equations PX = Q where P ∈ R^3 × 3 and Q ∈ R^3 × 1. Suppose P has an LU decomposition, P = LU, where L = bmatrix 1 & 0 & 0 \\ l_21 & 1 & 0 \\ l_31 & l_32 & 1 bmatrix U = bmatrix u_11 & u_12 & u_13 \\ 0 & u_22 & u_23 \\ 0 & 0 & u_33 bmatrix. Which of the following statement(s) is/are TRUE?

The system PX = Q can be solved by first solving LY = Q and then UX = Y.

If P is invertible, then both L and U are invertible.

If P is singular, then at least one of the diagonal elements of U is zero.

The system PX = Q can be solved by first solving LY = Q and then UX = Y.

If P is invertible, then both L and U are invertible.

If P is singular, then at least one of the diagonal elements of U is zero.

If P is symmetric, then both L and U are symmetric.

GATE CSE 2025 Session 2 Q44: True statements about the LU decomposition P = LU

Engineering Mathematics
2 marks
MSQ
No negative marking

Consider a system of linear equations $PX = Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 1}$ . Suppose $P$ has an LU decomposition, $P = LU$ , where

$L = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} \quad\text{and}\quad U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}.$

Which of the following statement(s) is/are TRUE?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, B, C (official GATE 2025 final answer key)

Concept used: $\det P = \det L \cdot \det U = u_{11}u_{22}u_{33}$ , and triangular systems solve by forward and back substitution.

First step: Use $\det L = 1$ to relate the invertibility of $P$ to the diagonal of $U$ ; a non-diagonal triangular matrix is never symmetric.

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE CSE 2025 Session 2 Q44: True statements about the LU decomposition P = LU

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