Matrices Watch

Scott3142
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#1
Report Thread starter 7 years ago
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Let A = \begin{pmatrix} 1 & 1 & 0 & ... & 0\\ 0 & 1 & ... & ... & ... \\ ... & ... & ... & ... & 0 \\ ... & ... & ... & 1 & 1 \\ 0 & ... & ... & 0 & 1\end{pmatrix} be an nxn matrix.

Find A^{-1} using elementary row operations.

I can see by just doing a few that the answer will be

A^{-1} = \begin{pmatrix} 1 & -1 & 1 & ... & 1\\ 0 & 1 & -1 & ... & ... \\ ... & ... & ... & ... & ... \\ 0 & ... & 1 & -1 & ... \\ ... & ... & ... & 1 & -1 \\ 0 & ... & ... & 0 & 1\end{pmatrix}.

However I think I should prove this somehow, induction perhaps. I have written A as

(A)_{ij}= \left\{ \begin{array}{ll}

		1  & \mbox{if } i\in \{j,j+1\} \\

		0 & \mbox{otherwise} 

	\end{array}

\right

And similarly

(A^{-1})_{ij} = \left\{ \begin{array}{ll}

		1  & \mbox{if } i=j \\

		-1 & \mbox{if } i=j+1 \\ 

               0 & \mbox{otherwise}                

	\end{array}

\right.

and tried to do something with that but I haven't got anywhere. Am I making more of it than is necessary?

Thanks.
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ghostwalker
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(Original post by Scott3142)
Thanks.
I suggest putting an example into Excel or whatever your favourite matrix multiplying software is; what you have there doesn't work.

Nor do your formulae match the matrices you've quoted at the top.
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Scott3142
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I have worked out the first 3 rows from the bottom of A^{-1} and it seems to be upper triangular with alternating 1s and minus 1s. I think this is right. Ahh I see how my formulae are wrong now but how do I generalise the operations to nxn?
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ghostwalker
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(Original post by Scott3142)
I have worked out the first 3 rows from the bottom of A^{-1} and it seems to be upper triangular with alternating 1s and minus 1s. I think this is right. Ahh I see how my formulae are wrong now but how do I generalise the operations to nxn?
What you are doing is starting from the nth row working up, subtracting the i'th row from the (i-1)'th row, which is generating your alternating pattern in the upper triangle.

If you look at it in terms of the augmented matrix (A|I); these operations reduce A to I, and the I to A^(-1). So just formalise that some way.

Alternatively you could take the approach of considering the elementary row operations as matrices in their own right, and you are generating a string of matrices that perform the reduction, and this allows you to state the inverse, but I'm not familiar with the nomenclature, etc. as it's donkey's years since I looked at this - and have never used it since..
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