## Find two 2-by-2 invertible matrices with sum not invertible

Linear spaces and subspaces, linear transformations, bases, etc.
testing
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### Find two 2-by-2 invertible matrices with sum not invertible

Find two 2-by-2 invertible matrices whose sum is not invertible.

Do I just start making up matrices, until I find something that works?

stapel_eliz
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testing wrote:Do I just start making up matrices, until I find something that works?

Yes and no. Random matrices may not be helpful. Instead, think about the relationship between invertible matrices and their determinants, and see if you can think of a way to use this to create a useful pair.

Eliz.

testing
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### Re: Find two 2-by-2 invertible matrices with sum not invertible

The determinant for a non-invertible matrix is zero, so these should work:

$\left[\begin{array}{rr}1&0\\0&1\end{array}\right]$

$\left[\begin{array}{rr}-1&0\\0&-1\end{array}\right]$

If you add them, you get:

$\left[\begin{array}{rr}0&0\\0&0\end{array}\right]$

And any two matrices like that will work, so that's loads of answers right there.

stapel_eliz
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Exactly!