Consider a system of linear equations PX = Q where P R3 3 and Q R3 1. Suppose P has an LU decomposition, P = LU, where L = bmatrix 1 & 0 & 0 \\ l21 & 1 & 0 \\ l31 & l32 & 1 bmatrix and U = bmatrix u11 & u12 & u13 \\ 0 & u22 & u23 \\ 0 & 0 & u33 bmatrix. Which of the following statement(s) is/are TRUE? A) The system PX = Q can be solved by first solving LY = Q and then UX = Y. B) If P is invertible, then both L and U are invertible. C) If P is singular, then at least one of the diagonal elements of U is zero. D) If P is symmetric, then both L and U are symmetric.

Official answer: A) The system PX = Q can be solved by first solving LY = Q and then UX = Y.; B) If P is invertible, then both L and U are invertible.; C) If P is singular, then at least one of the diagonal elements of U is zero.. Explanation: Let us analyze each statement based on the properties of LU decomposition: - Statement (A): Substituting P = LU into PX = Q gives L(UX) = Q. Letting Y = UX, we first solve LY = Q for Y by forward substitution, and subsequently solve UX = Y for X by back substitution. Thus, statement (A) is TRUE. - Statement (B): The determinant of the product of two matrices is the product of their determinants: (P) = (LU) = (L) (U) Since L is a unit lower triangular matrix, (L) = 1 0, meaning L is always invertible. If P is invertible, then (P) 0, which implies: (U) = (P) (L) = (P) 0 Thus, U is also invertible. Therefore, statement (B) is TRUE. - Statement (C): If P is singular, then (P) = 0. Since (L) = 1, we have: (P) = (L) (U) = 1 (u11 u22 u33) = 0 This requires at least one of the diagonal elements of U (u11, u22, or u33) to be zero. Thus, statement (C) is TRUE. - Statement (D): A triangular matrix (upper or lower) can only be symmetric if it is a diagonal matrix. For a general symmetric matrix P, L has non-zero off-diagonal entries below the diagonal and U has non-zero off-diagonal entries above the diagonal, so neither L nor U is symmetric in general (instead, P = LDLT). Thus, statement (D) is FALSE. Hence, the correct options are A, B, and C.

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

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

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

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

Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 2 Q44 · MSQ · 2 marks

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 $L = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} \text{ and } U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}.$ Which of the following statement(s) is/are TRUE?

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 2 Q44 · MSQ · 2 marks

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 $L = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} \text{ and } U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}.$ Which of the following statement(s) is/are TRUE?

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 2

More on Linear Algebra

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 2

More on Linear Algebra

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 2

More on Linear Algebra

Share this solution

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 2

More on Linear Algebra

Share this solution