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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