Diagonalize

construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.

Just write down a diagonal matrix with one zero on the diagonal and then apply an orthogonal transformation. E.g. if you start with the matrix:

A =
[1 ,0
0,1]

And take the orthogonal transformation to be:

S =
[cos(theta) , -sin(theta)
sin(theta), cos(theta]

then

the transformed matrix is:

S A S^(-1)

S^(-1) =
[cos(theta) , sin(theta)
-sin(theta), cos(theta]

If you take theta = pi/4 you get the matrix:

1/2 [1,1
1,1]

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