Question

Research the Cayley-Hamilton Theorem and how it can be used to compute the inverse of a non-singular square matrix.

Answer 1

By Cayley-Hamilton Theorem matrix is a root of its characteristic polynomial, so

$$A ^ {n} + c _ {1} A ^ {n - 1} + \dots + c _ {n - 1} A + c _ {n} = 0$$

As it is known, $c_{1} = \operatorname{tr}(A), c_{n} = \det A$ . If $A$ is non-singular, then $\det A \neq 0$ and

$$\left(A ^ {n - 1} + c _ {1} A ^ {n - 2} + \dots + c _ {n - 1}\right) A = - \det A$$ $$\left(- \frac {1}{\det A} A ^ {n - 1} - \frac {c _ {1}}{\det A} A ^ {n - 2} - \dots - \frac {c _ {n - 1}}{\det A}\right) A = 1 _ {n}$$

Thus $-\frac{1}{\det A} A^{n - 1} - \frac{c_1}{\det A} A^{n - 2} - \dots -\frac{c_{n - 1}}{\det A}$ will be inverse for A.