Consider a system of linear equations \(𝑃𝑋 = 𝑄\) where \(𝑃 ∈ ℝ^{3×3}\) and…

GATE · 2025 · CS · Set 2 · Computer Science & IT

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

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

1. A.

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

2. B.

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

3. C.

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

4. D.

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

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