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September 27th, 2008, 08:43 AM   #1
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Finding values for non singular matrix

Dear Math Forum, regarding the 3x3 matrix shown below. I need to find out for which values of a the matrix is non singular. I have tried reducing it to figure out if the set is linearly independent but I can't seem to get it right. Can you help?

1 1 -1
0 1 2
1 1 a

Best regards,
Aske

PS. I am new in matrix calculus (1'st year at college) so sorry if the question is too simple.
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September 27th, 2008, 09:02 AM   #2
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Re: Finding values for non singular matrix

A matrix M is non-singular if its determinant is non-zero. Do you know how to find the determinant of a 3x3 matrix?
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September 27th, 2008, 10:04 AM   #3
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Re: Finding values for non singular matrix

I know how to find the determinant for a 3x3 matrix. According to the assignment we all ready know that the matrix is non-singular but I need to find out for which values of "a" it is non-singular. It is to be said that we haven't yet been through determinants of 3x3 matrices in our curriculum so there must be another way to solve it.
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September 27th, 2008, 10:21 AM   #4
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Re: Finding values for non singular matrix

If you have covered elementary row operations, it is very simple to convert your matrix into an upper triangular matrix, whose determinant - the trace - will be the same as the original matrix.
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September 27th, 2008, 03:14 PM   #5
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Re: Finding values for non singular matrix

Otherwise, think about linear systems:
For any matrix A, Ax=0 has only the trivial solution of x if and only if it is non-singular; so do basic row operations to get it into row echelon form, and you want the last row to be no-zero, that way z must be zero, meaning only one solution. You can also set it up as an augmented matrix with the last column being all zeroes, but that column won't do anything:


(I just subtracted the first row from the third).

If a=-1, the last row is the 0=0 identity, which means we need to pick a parameter for our third variable -> infinite solutions -> singular. So any a?-1 will give a non-singular matrix.
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