Question-8

Let \(M_{2 \times 2}(\mathbb{R})\) represent the set of all \(2 \times 2\) real matrices. Now consider the following sets: \[ \begin{aligned} V_1 &= \{A\ |\ A \in M_{2 \times 2}(\mathbb{R}) \text{ and } A \text{ is a diagonal matrix}\}\\ V_2 &= \{A\ |\ A \in M_{2 \times 2}(\mathbb{R}) \text{ and sum of diagonal elements of } A \text{ is equal to 1}\} \end{aligned} \] Which of the following statements is true?

• \(V_1\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_2\) is not

• \(V_2\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_1\) is not

• Both \(V_1\) and \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

• Neither \(V_1\) nor \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

Hint

Check for the zero vector.

Answer

• \(V_1\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_2\) is not

• \(V_2\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_1\) is not

• Both \(V_1\) and \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

• Neither \(V_1\) nor \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

Solution

\(V_2\) does not have the zero element. \(V_1\) has the zero element, is closed under addition and scalar multiplication.