Finding the inverse of matrices

matt grime

So, find the inverse of 17 mod 26, it's no harder than doing Euclid's algorthim (or being clever and just checking my list on the last page)

( from scratch, less than one minutes thinking gives 23 as the inverse, or -3, equivalently)

cronxeh

I tried coming up with general number of operations you'd perform with either method to determine which is faster, and I just dont have time right now..

matt, which one is faster - augmented matrix or adjoint/determinant method? Ive counted about 47 operations on adj/det method, and I'm guessing there is more in augmented method

ComputerGeek

Quote by Hurkyl

One thing that's sure to work (though it might not be the easiest) is to compute the adjoint (I think that's the word used) -- the matrix whose (i,j)-th entry is the (i,j)-th subdeterminant. The identity A * adj A = (det A) I will yield the inverse.

Another approach is to use the chinese remainder theorem. Work out the inverse mod 13 and mod 2, then combine them to get the mod 26 answer.

All that said, you probably made an arithmetic error -- I don't think that last row is correct. (Because if that 22 is right, all of the terms on the RHS should be even)

the second option might be best because then every element on mod 2 and mod 13 will be a unit and you can find an inverse for every number. mod 26 is harder because not every element will have an inverse and you may need to do some careful row switches.

ComputerGeek

Quote by Hurkyl

Yep, but then again, the matrix is invertible if and only if the determinant is invertible.

don't you mean not equal to zero?

Data

Not equal to zero and invertible mean the same thing for elements of a field~

matt grime

But we aren't dealing with a field. Invertible is indeed the correct term.

Hurkyl

For example, an integer matrix has an integer inverse if and only if its determinant is 1 or -1 --- because those are the only invertible integers!

ComputerGeek

ah... that makes sense.

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matt grime Recognitions: Homework Help Science Advisor	Finding the inverse of matrices So, find the inverse of 17 mod 26, it's no harder than doing Euclid's algorthim (or being clever and just checking my list on the last page) ( from scratch, less than one minutes thinking gives 23 as the inverse, or -3, equivalently)

cronxeh Recognitions: Gold Member	I tried coming up with general number of operations you'd perform with either method to determine which is faster, and I just dont have time right now.. matt, which one is faster - augmented matrix or adjoint/determinant method? Ive counted about 47 operations on adj/det method, and I'm guessing there is more in augmented method

Data	Not equal to zero and invertible mean the same thing for elements of a field~

Hurkyl Recognitions: Gold Member Science Advisor Staff Emeritus	For example, an integer matrix has an integer inverse if and only if its determinant is 1 or -1 --- because those are the only invertible integers!