how to find inverse of a matrix

Question

How to find the inverse of a 4x4 order matrix using adjoints for example

$$A=\begin{pmatrix} 2 & -6 & -2 & -3 \\ 5 &-13 &-4 &-7 \\ -1 & 4& 1& 2 \\ 0 & 1 & 0 & 1 \\\end{pmatrix}$$

Answer 1

The adjoint of a square matrix $A$ is the transpose of the cofactor matrix $C$. The minor of the entry of $A$ in row $i$ column $j,$ denoted $M_{i,j}$ is the determinant of the submatrix formed by deleting the $i$-th row and $j$-th column. Then, the $(i,j)$ entry of C is simply defined as $(-1)^{i+j}\cdot M_{i,j}$.

For the above matrix, you get the following cofactor matrix: $$C=\begin{pmatrix} 2 & -1 & 4 & 1 \\ -1 & 0 & -1 & 0 \\ 0 & -2 & 3 & 2 \\ -1 & 1 & -1 & -2 \\\end{pmatrix}$$

Taking the transpose, you get the adjoint: $$Adj(A)=\begin{pmatrix} 2 & -1 & 0 & -1 \\ -1 & 0 & -2 & 1 \\ 4 & -1 & 3 & -1 \\ 1 & 0 & 2 & -2 \\\end{pmatrix}$$

To get $A^{-1}$ from $Adj(A)$ you have to scale by the (inverse of the) determinant of $A$. You don't actually need to explicitly compute $det(A)$ in this case since you can check that $Adj(A)\cdot A = -I$, so you see that it must be $det(A) = -1$, and $A^{-1}$ is just $Adj(A)$ with entries with opposite sign.