^^

Thank you very much!

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- Dec 7th 2008, 03:33 AMishanProve that a matrix in NOT invertible if an eignvalue is 0
^^

Thank you very much! - Dec 7th 2008, 03:44 AMMoo
In the basis that diagonalizes the matrix, the matrix can be written :

Where are the eigenvalues of the matrix.

The determinant in this base is obviously the product of the

So if one of the eigenvalues is 0, the determinant is 0.

And since the determinant doesn't change with the basis, we have proved that det A = 0, and hence it is not invertible.

Edit : waaaah 39:):)th post :p - Dec 7th 2008, 04:01 AMwhipflip15
The determinant of a matrix is equal to the product of the eigenvalues. Hence the determinant is 0 and the matrix is not invertible.

- Dec 7th 2008, 09:18 AMThePerfectHacker
If has as an eigenvalue then it means for .

Therefore, for .

This means is not invertible.