# Matrix Inverse

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Can someone please shed some light on getting the inverse of a matrix(4x4)....i am not totally clear on it and i have hit google hard but i cant make heads nor tails of it...can someone dumb it down a notch

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If your 4x4 matrix is just a rotation+translation, then there's a big shortcut - e.g. see

http://graphics.stanford.edu/courses/cs248-98-fall/Final/q4.html

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Probably the easiest way to get the inverse (at least to visualize) is to start with a 4x8 matrix, with the 4x4 matrix you want to invert on the left hand side, and the 4x4 identity matrix on the right hand side like:
[ a b c d 1 0 0 0 ][ e f g h 0 1 0 0 ][ i j k l 0 0 1 0 ][ m n o p 0 0 0 1 ]

From there, you do row operations (swaps, row additions and scalar multiplication) to try to get the left hand side to look like the identity matrix. For example, if a is non-zero, you might divide the first row by a to get:
[ 1 b' c' d' 1/a 0 0 0 ][ e f  g  h  0   1 0 0 ][ i j  k  l  0   0 1 0 ][ m n  o  p  0   0 0 1 ]

Then subtract e times the first row from the second row to get:
[ 1 b' c' d'  1/a 0 0 0 ][ 0 f' g' h' -e/a 1 0 0 ][ i j  k  l  0    0 1 0 ][ m n  o  p  0    0 0 1 ]

Do the same thing with the third and fourth rows to get:
[ 1 b' c' d'  1/a 0 0 0 ][ 0 f' g' h' -e/a 1 0 0 ][ 0 j' k' l' -i/a 0 1 0 ][ 0 n' o' p' -m/a 0 0 1 ]

Now try to get the second column to look like 0 1 0 0, so you might divide the second row by f' and subtract multiples of the second row from all the other rows. Eventually you'll end up with something that looks like:
[ 1 0 0 0 a'' b'' c'' d'' ][ 0 1 0 0 e'' f'' g'' h'' ][ 0 0 1 0 i'' j'' k'' l'' ][ 0 0 0 1 m'' n'' o'' p'' ]

And the right hand side will be the inverse you were trying to find.

There are a couple of things to watch out for if you do this. Sometimes you'll have a zero in the diagonal where you're trying to get a 1. In that case, swap the row with another lower row where it doesn't have a 0 there. If there isn't a row that doesn't have a 0, then the matrix is non-invertible. Also, you can get some bad rounding problems using a straightforward implementation of this technique, so it's not recommended for general use in finding matrix inverses.

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can some explain CoFactor and how it would come into play here?

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For this technique of determining the matrix inverse, cofactors aren't used to determine the solution.

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I believe CoFactors come into play with the Cramer's Rule approach, which is a poor approach----extremely expensive if the matrix is larger than 4x4.

Can you shed some light on exactly why you need the inverse? The game development uses of the inverse should be mostly well covered, even in the forum archives here (which are searchable).

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I have exactly what you need...

you have "m" arranged as thus:
m =
[ m11, m12, m13, m14]
[ m21, m22, m23, m24]
[ m31, m32, m33, m34]
[ m41, m42, m43, m44]

then its inverse is the following

inverse( m ) =
[ m22*m33*m44-m22*m34*m43-m32*m23*m44+m32*m24*m43+m42*m23*m34-m42*m24*m33, -m12*m33*m44+m12*m34*m43+m32*m13*m44-m32*m14*m43-m42*m13*m34+m42*m14*m33, m12*m23*m44-m12*m24*m43-m22*m13*m44+m22*m14*m43+m42*m13*m24-m42*m14*m23, -m12*m23*m34+m12*m24*m33+m22*m13*m34-m22*m14*m33-m32*m13*m24+m32*m14*m23]
[ -m21*m33*m44+m21*m34*m43+m31*m23*m44-m31*m24*m43-m41*m23*m34+m41*m24*m33, m11*m33*m44-m11*m34*m43-m31*m13*m44+m31*m14*m43+m41*m13*m34-m41*m14*m33, -m11*m23*m44+m11*m24*m43+m21*m13*m44-m21*m14*m43-m41*m13*m24+m41*m14*m23, m11*m23*m34-m11*m24*m33-m21*m13*m34+m21*m14*m33+m31*m13*m24-m31*m14*m23]
[ m21*m32*m44-m21*m34*m42-m31*m22*m44+m31*m24*m42+m41*m22*m34-m41*m24*m32, -m11*m32*m44+m11*m34*m42+m31*m12*m44-m31*m14*m42-m41*m12*m34+m41*m14*m32, m11*m22*m44-m11*m24*m42-m21*m12*m44+m21*m14*m42+m41*m12*m24-m41*m14*m22, -m11*m22*m34+m11*m24*m32+m21*m12*m34-m21*m14*m32-m31*m12*m24+m31*m14*m22]
[ -m21*m32*m43+m21*m33*m42+m31*m22*m43-m31*m23*m42-m41*m22*m33+m41*m23*m32, m11*m32*m43-m11*m33*m42-m31*m12*m43+m31*m13*m42+m41*m12*m33-m41*m13*m32, -m11*m22*m43+m11*m23*m42+m21*m12*m43-m21*m13*m42-m41*m12*m23+m41*m13*m22, m11*m22*m33-m11*m23*m32-m21*m12*m33+m21*m13*m32+m31*m12*m23-m31*m13*m22]

as much as it appears ugly and unintuitive, it is by far faster than any other method especially if you spot recurrent muls and assign them into variables...

general methods however should be considered when working with other than 4x4 matrices (uncommon with 3d game programming however)

enjoy

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There you can find code for 4x4 matrix inverse using several different methods: Clickster
Gaussian elimination is faster for 5x5 matrices and bigger.

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Quote:
 Original post by BrainCodingArtsI have exactly what you need...you have "m" arranged as thus:m =[ m11, m12, m13, m14][ m21, m22, m23, m24][ m31, m32, m33, m34][ m41, m42, m43, m44]then its inverse is the following...as much as it appears ugly and unintuitive, it is by far faster than any other method especially if you spot recurrent muls and assign them into variables...general methods however should be considered when working with other than 4x4 matrices (uncommon with 3d game programming however)enjoy

You need to divide the elements of the matrix by the determinant of m after that big block of math, or if the determinant comes out to be 0 (do this before finding the adjugate), just skip out or throw an exception or something because it is not invertable. What you have there is the adjugate, which is the transpose of the matrix of the cofactors (phew ;))

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Quote:
 Original post by BrainCodingArtsas much as it appears ugly and unintuitive, it is by far faster than any other method especially if you spot recurrent muls and assign them into variables...

Actually, that method is not faster than other methods. In fact, its the slowest method. This appears to be a fragment of the Cramer's rule method. 4x4 is absolutely the largest matrix you would EVER want to apply Cramer's rule to. This is an n4 operation. Yes, you read correctly, its an n-to-the-FOURTH operation. That means a 4x4 matrix inverse requires O(256) multiply/divide/add/subtract operations (count them in your equation). In contrast, naive Gaussian Elimination, a n3 operation is far cheaper, requiring only O(64) operations. I only point this out in the interest of education, :).

I will give you this. For some types of matrices (symmetric matrices) there can be recurrent operations that can be cached and reused, thus reducing the expense of Cramer's rule. But, the general fact is that it is by far the slowest technique. Might as well code up one of the other methods.

The following web page has a more in depth discussion. Note especially the table comparing the expansion by minors bit of Cramer's rule to Gaussian Elimination:

Operation Counts

[Edited by - grhodes_at_work on December 27, 2004 1:57:32 PM]

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