# 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 :

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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.