Calculating the inverse of a 3x3 matrix by hand is a tedious job. But this has several uses, including solving various matrix equations.
1Find det(M), the determinant of the Matrix M. The determinant will usually show up in the denominator of the inverse. If the determinant is zero, the matrix won't have an inverse.Ad
2Find MT, the transpose of the matrix. Transposing means reflecting the matrix about the main diagonal, or equivalently, swapping the (i,j)th element and the (j,i)th.
3Find the determinant of each of the 2x2 minor matrices.Find the determinant of each of the 2x2 minor matrices.
4Represent these as a matrix of cofactors as shown, and multiply each term by the sign indicated. The result of these steps is the adjugate matrix (sometimes also called the adjoint), notated Adj(M).
5Find the inverse by dividing the adjugate found in the previous step by the determinate from the first step.
6One can combine these steps by transposing, copying over the first two columns and rows, and finding the 2x2 determinants around each dot. Checking one's work computes the determinant three times; if they agree, this is the denominator. With this "torus" method, the signs are already correct.Ad
Are you an expert on naturally whitening skin?
If so, share what you know about how to naturally whiten skin at home.
Thanks for sharing your knowledge.
We are a bit smarter today because of you.
- Note that this same method can be applied to a matrix containing variables or unknowns, for example an algebraic matrix, M, and its inverse, M-1.
- Write down all your steps as it is extremely difficult to inverse a 3x3 matrix in your head
- Computer programs exist that work out the inverses of matricies for you, up to and including the size of 30x30 matricies
- The adjugate matrix is the transpose of the matrix of cofactors, that is why we transpose the matrix in step 2, to find the a transposed matrix of cofactors.
- Check that is accurate by multiplying M by M-1. You should be able verify that M*M-1 = M-1*M = I. I is the identity matrix, consisting of 1s along the main diagonal and 0s elsewhere. If not, you made an error somewhere.
- Not all 3x3 matrices have inverses. If the determinant of the matrix is equal to 0, then it does not have an inverse. (Notice that in the formula we divide by det(M). Division by zero is not defined.)
Sources and Citations
Recent edits by: Goatee123, Jamie, Syzygies
In other languages:
Deutsch: Wie man eine 3x3 Matrix umkehrt, Español: Cómo invertir una matriz de 3X3, Français: Comment calculer l'inverse d'une matrice 3x3, Italiano: Come Invertire una Matrice 3X3, Português: Como Inverter uma Matriz de 3X3, Русский: найти обратную матрицу 3х3, 中文: 求3X3矩阵的逆矩阵, Nederlands: De inverse van een 3x3 matrix bepalen
Thanks to all authors for creating a page that has been read 1,251,637 times.