# What is the derivation for formula of inverse matrices (3x3 matrix) - algebraically derive the formula?

I have derived the formula for inversing a $$2\times 2$$ matrix.

You can access the pages where i have derived the formula for inverse of $$2\times 2$$ matrix :

Page 1

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Those who don't want to mess with the page links above can use the below linked video : I used this video as a guide to derive the formula for inverse of $$2\times 2$$ matrix - Determining the Inverse of a Matrix Algebraically

The formula goes like this : $$A^{-1} = \dfrac{1}{\det(A)}\cdot\operatorname{adj} (A)$$

As i try to derive the formula for inversing a $$3\times 3$$ matrix, it became exponentially difficult. Can anyone please guide me with the derivation?

Thanks a lot! Hope this will help many who love Matrices :)

• The formula for the inverse of a $2 \times 2$ matrix can also be found quickly by guessing. I think it's not very important or useful to have a formula for the inverse of a $3 \times 3$ matrix. To compute the inverse of any specific $n \times n$ matrix, one can solve $AX = I$ using Gaussian elimination. Cramer's rule is also worth knowing about, and very nice / simple proofs of Cramer's rule can be found on math.stackexchange. Jun 8 '19 at 7:26
• @littleO Thank you for getting this to my attention. I'll surely check out Gaussian elimination & Cramer's rule. Jun 8 '19 at 7:30
• @littleO I had a look at Gaussian elimination and all. It looks like they're giving a formula upright. But i just need to know how they came up with that formula (definition for that formula). Is there anything like that ? Jun 8 '19 at 9:38
• When you say "it looks like they're giving a formula upright", which formula are you referring to? Gaussian elimination is not a formula, it's an algorithm that you can use to solve a linear system of equations. Jun 8 '19 at 9:48
• I think this is a nice proof of Cramer's rule: math.stackexchange.com/a/1941606/40119 For Gaussian elimination, if we have a system of linear equations, we know that performing any of the following operations will not change the set of solutions to the linear system: 1) multiplying both sides of any one of the equations by a nonzero constant; 2) adding one of the equations to another one of the equations; 3) interchanging two equations. Gaussian elimination is just a systematic way of performing these operations in order to simplify the linear system, until the solution is clear. Jun 8 '19 at 10:18

Just find the cofactor matrix, then the adjunct matrix is the transpose of that. Divide the adjunct by the determinant of the original matrix, e.g.

$$A^{-1}=\frac{1}{\det A}C^\text{T},$$

where $$C$$ is the cofactor matrix and $$\det A\neq 0$$.

Since you are a developer, and if you're trying to implement this, you may want to use an existing matrix library as many of them will be optimized for efficiency, e.g. using LU decomposition.

You can always find the general inverse of any square matrix by augmenting it with the identity matrix, then use Gaussian elimination (putting it into reduced row echelon form), so that you get the identity on the left. Whats on the right will be the inverse.

You may find this link useful.

• Firstly, thanks for your answer. But, i need a derivation not a magic formula. And i asked this question from a Mathematician viewpoint & not from a Developer viewpoint. There is no way for google to ask for your email id when you open the pages (i tried to open the page links from incognito window & it worked). Jun 8 '19 at 7:09
• Doesn't do the same for me so there certainly is a way for google to ask ! But maybe that's because I'm anroad at the moment. As mentioned you could look at an LU decomposition approach, or else use the reduced row echelon form I just added. Hope it helps a little. Jun 8 '19 at 7:19
• You can use the video that i linked in the question if you can't access the pages. Thank you for your effort and time. :) Jun 8 '19 at 7:21
• The guy in the video is using Gaussisn elimination. Try to persevere with it bearing in mind you should be able to factor out the reciprocal of the determinant. Jun 8 '19 at 7:26
• Thanks for the link! Jun 8 '19 at 7:32