## Linear Algebra and Its Applications, Exercise 2.1.9

Exercise 2.1.9. Consider the set of all nonsingular 2 by 2 matrices. Is it a vector space? How about the set of singular 2 by 2 matrices?

Answer: The set of all nonsingular 2 by 2 matrices is not closed under addition; for example, the matrices $I = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} \qquad P = \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$

are both nonsingular (and in fact are their own inverses), but the sum $I + P = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} + \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix} = \begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}$

is singular. Since the set of nonsingular 2 by 2 matrices is not closed under addition it is not a vector space.

Next, consider the two singular matrices $A = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \qquad B = \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}$

Their sum is nonsingular: $A + B = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} + \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix} = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} = I$

So the set of singular  2 by 2 matrices is also not closed under addition, and thus is also not a vector space.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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