I know that:
$\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1}=\frac{1}{Determinant} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
But what is the equivalent expression for 3x3 (or higher dimensional matrices)?
Thanks
I know that:
$\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1}=\frac{1}{Determinant} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
But what is the equivalent expression for 3x3 (or higher dimensional matrices)?
Thanks
The best way to find inverses is by Gaussian-Jordan elimination. Because that is the most efficient algorithm. However, it is not pleasant. There is a very elegant way to find inverses but when matrices get large it because computationally difficult to do (still it has important theoretical uses).
Step 1: Given $\displaystyle A$ a $\displaystyle n\times n$ matrix compute $\displaystyle \det A$. If this determinant is zero then it have no inverse.
Step 2: Compute the cofactor matrix. Meaning for each $\displaystyle a_{ij}$ entry in the matrix compute the cofactor of $\displaystyle a_{ij}$. Replace each entry by its cofactor. This will form a matrix called the cofactor matrix.
Step 3: Compute the adjoint matrix. To do this find the transpose of the cofactor matrix. The transpose operation on $\displaystyle M$ (denoted by $\displaystyle M^T$) is flipping the elements along the main diagnol. So for the $\displaystyle a_{ij}$ entry replace by $\displaystyle a_{ji}$ entry. (Note, that the main diagnol of the matrix is unchanged by transpose operation).
Step 4: Now divide each element in the adjoint matrix by the determinant. The resulting matrix is the inverse.
I do not own any of those new fancy show-off calculators. But I remeber from playing around with one of the TI's that I found a way to find inverse of matrices. There is a matrix command bottom. Program your matrix then press in the matrix into the screen followed by $\displaystyle \boxed{^{-1}}$ and then enter. This will give you a matrix which is the inverse.