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Question 4.1.14: Consider the set W of all noninvertible 2 × 2 matrices. Is W......

Consider the set W of all noninvertible 2 × 2 matrices. Is W a subspace of ℝ^{2×2}?

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The following example shows that W isn’t closed under addition:

\begin{bmatrix}1&0\\0&0\end{bmatrix}+\begin{bmatrix}0&0\\0&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.

\begin{matrix}& \ \ \ \nwarrow\nearrow&\\&\text{in W}&\end{matrix}          \begin{matrix}\uparrow\\ \text{not in W}\end{matrix}

Therefore, W fails to be a subspace of ℝ^{2×2}.

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