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November 9th, 2018, 08:16 PM   #1
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Inverse of a matrix of decimal or floating numbers.

Can we find the Inverse of a 3*3 matrix having
the following decimal number elements?

1.2 2.3 4.5
6.1 4.2 2.9
7.4 3.5 8.7

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Prashant S Akerkar
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November 9th, 2018, 08:23 PM   #2
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https://matrix.reshish.com/inverse.php
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November 9th, 2018, 09:35 PM   #3
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$\begin{pmatrix}
-2639/8482 & 426/8482 & 1223/8482 \\
3161/8482 & 2286/8482 & -2397/8482 \\
973/8482 & -1282/8482 & 899/8482
\end{pmatrix}$
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November 9th, 2018, 10:00 PM   #4
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Quote:
Originally Posted by akerkarprash View Post
Can we find the Inverse of a 3*3 matrix having
the following decimal number elements?

1.2 2.3 4.5
6.1 4.2 2.9
7.4 3.5 8.7

Thanks & Regards,
Prashant S Akerkar
Multiply your matrix elements by 10 and divide by 10 on the "outside" of the matrix. Then you have whole numbers to work with. (If that's the issue.)

-Dan
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November 10th, 2018, 02:08 AM   #5
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Thanks.

Can it become complex to compute the
Inverse if the precision is increased
for the decimal numbers in the Matrix?

Example:

34.333333333 51.222222785 98.333556677
12.555555555 76.444555532 65.234567879
34.888855338 68.542666669 54.236666856

Thanks & Regards,
Prashant S Akerkar
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November 10th, 2018, 02:04 PM   #6
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Originally Posted by akerkarprash View Post
Thanks.

Can it become complex to compute the
Inverse if the precision is increased
for the decimal numbers in the Matrix?

Example:

34.333333333 51.222222785 98.333556677
12.555555555 76.444555532 65.234567879
34.888855338 68.542666669 54.236666856

Thanks & Regards,
Prashant S Akerkar
Yes, my idea won't work well that way. Still, I presume you are trying to find the inverse matrix using row operations? There is no problem using decimals for that. It's just a touch uglier.

-Dan
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November 10th, 2018, 03:00 PM   #7
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If the numbers in the matrix are known to be approximate, you have a difficulty if the determinant of the matrix seems to be zero (or very close to zero).
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November 10th, 2018, 03:16 PM   #8
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Quote:
Originally Posted by skipjack View Post
If the numbers in the matrix are known to be approximate, you have a difficulty if the determinant of the matrix seems to be zero (or very close to zero).
Very common misunderstanding! It is not a problem if the determinant is close to 0. It is the condition number that matters, not the determinant. Contrary to expectation, a small determinant does not indicate near-invertibility, nor it is a problem.
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November 11th, 2018, 02:39 AM   #9
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$\begin{pmatrix}
34.333333333 & 51.222222222 & 98.333333333 \\
12.555555555 & 76.444444444 & 65.222222222 \\
21.777777778 & -25.222222222 & 33.111111111
\end{pmatrix}$

The above matrix, where the first two rows are similar to the corresponding rows in the matrix supplied by akerkarprash, but the third row is the difference of the first two rows, so that the determinant of the matrix is zero, doesn't have an inverse. Nevertheless, the website mentioned above calculates an inverse for it (as does wolframalpha). Of course the condition number is infinite in this case.

If I change the final decimal place of one of the entries in this matrix, the determinant of the matrix is very small and the above website calculates its inverse inaccurately, but the condition number is large.

For a matrix of this type, you are effectively asserting that its determinant can be very close to zero without its condition number being very large. Can you give an example of this?
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