Inverse of matrix using Cayley Hamilton method Example [[5,1],[4,2]]

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Home > Matrix & Vector calculators > Inverse of matrix using Cayley Hamilton method example

3. Cayley Hamilton method example ( Enter your problem )

( Enter your problem )

Other related methods

3. Example

$[[2,3],[4,10]]$
(Previous example)

4. Example $[[5,1],[4,2]]$

Find Inverse of matrix using Cayley Hamilton method
$A=[[5,1],[4,2]]$

Solution:
To apply the Cayley-Hamilton theorem, we first determine the characteristic polynomial p(t) of the matrix A.

$|A-tI|$

=

	$(5-t)$	$1$
	$4$	$(2-t)$

$=(5-t) × (2-t) - 1 × 4$

$=(10-7t+t^2)-4$

$=t^2-7t+6$

$p(t)=t^2-7t+6$

The Cayley-Hamilton theorem yields that

$O = p(A)=A^2-7A+6I$

Rearranging terms, we have

$:. -6I = A(A-7I)$

$:. A^-1 = 1/-6(A-7I)$

Now, first we find

$A-7I$

$7 × I$

=

$7$

×

	$1$	$0$
	$0$	$1$

=

	$7$	$0$
	$0$	$7$

$A - 7 × I$

=

	$5$	$1$
	$4$	$2$

-

	$7$	$0$
	$0$	$7$

=

	$5-7$	$1$
	$4$	$2-7$

=

	$-2$	$1$
	$4$	$-5$

Now,

$A^-1 = 1/-6(A-7I)$

$:. A^-1 =$

$1/(-6)$

	$-2$	$1$
	$4$	$-5$

This material is intended as a summary. Use your textbook for detail explanation.
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