Answered: Show or explain why the set of all 2x2…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter6: Linear Systems

Section6.3: Matrix Algebra

Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...

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Question

Show or explain why the set of all 2x2 invertible matrices with the standard or usual matrix addition and scalar multiplication is not closed under scalar multiplication.

Be complete in setting up any information needed for your proof, executing the proof, and make sure your conclusion is clearly stated and the justification is understandable by an average reader of math from our class. Your work should flow in a logical progression of ideas.

Expert Solution

Step 1

The set of all invertible 2x2 matrices is not a vector space.

first point :

$\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}$ is invertible and $\begin{bmatrix} -1 &0 \\ 0 & -1 \end{bmatrix}$ is invertible, but $\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}$ + $\begin{bmatrix} -1 &0 \\ 0 & -1 \end{bmatrix}$ = $\begin{bmatrix} 0 &0 \\ 0 & 0 \end{bmatrix}$ is not invertible.

so it's not closed under addition.

second point:

I don't believe it's closed under scalar multiplication.
If k = 0, then k[a b, c d] = 0[a b, c d] = [0 0, 0 0]. Since det = 0, the matrix is not invertible.