Verify the Cayley Hamilton Theorem for A The Cayley Hamilton Theorem can be used to calculate powers and inverses of matrices For example, if A is a 2 Atilde 2 matrix with characteristic polynomial c A ( Icirc raquo ) Icirc raquo 2 a Icirc raquo b, then A 2 aA bI O, so It is easy to see that by continuing in this fashion we can express any positive power of A as a linear combination of I and A From A 2 aA bI O, we also obtain A(A aI ) bI, so provided b 0 1 2 0 A aA bI A AA A( aA bl) aA bA and ( ) b bl) bA (a b)A abI I

Question: Verify the Cayley-Hamilton Theorem for A = The Cayley-Hamilton Theorem can be used to calculate powers and inverses of matrices. For example, if A is

Verify the Cayley-Hamilton Theorem for A =
-1 -2 0.

The Cayley-Hamilton Theorem can be used to calculate powers and inverses of matrices. For example, if A is a 2 Ã— 2 matrix with characteristic polynomial c_A(Î») = Î»² + aÎ» + b, then A² + aA + bI = O, so

A? = - aA – bI A = AA? = A(-aA – bl) = -aA? – bA and — -а(-аА — ы) — bА bl) – bA (a – b)A + abI I||

It is easy to see that by continuing in this fashion we can express any positive power of A as a linear combination of I and A. From A² + aA + bI = O, we also obtain A(A + aI ) = -bI, so

-1 -2 0. A? = - aA bI A = AA? =

provided b ‰ 0.

-1 -2 0. A? = - aA bI A = AA? = A(-aA bl) = -aA? bA and -(- ) b bl) bA (a b)A + abI I|||

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