GATE CS 2001 Q1.1 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Consider the following statements:

Which one of the following statements is correct?

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Answer: A

We can determine the truth value of both statements by constructing simple examples using $2 \times 2$ matrices ( $n = 2$ ).

Statement S1: The sum of two singular $n \times n$ matrices may be non-singular.

Let us consider two singular matrices $A$ and $B$ :

A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

Both $A$ and $B$ are singular because $\det(A) = 0$ and $\det(B) = 0$ .

Their sum is:

A + B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I

Since $\det(I) = 1 \neq 0$ , the sum $A + B$ is non-singular. Thus, S1 is true.

Statement S2: The sum of two $n \times n$ non-singular matrices may be singular.

Let us consider two non-singular matrices $C$ and $D$ :

C = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \quad \text{and} \quad D = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -I

Both $C$ and $D$ are non-singular because $\det(C) = 1 \neq 0$ and $\det(D) = 1 \neq 0$ .

Their sum is:

C + D = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O

Since $\det(O) = 0$ , the sum $C + D$ is singular. Thus, S2 is true.

Since both $S1$ and $S2$ are true, the correct option is A.

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Published 10 Oct 2026

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