If [A = ( pmatrix 2 & 2 9 & 4 pmatrix )] and [I = ( pmatrix 1 & 0 0 & 1 pmatrix )] then 10…

Question

If $$A = \left( {\begin{pmatrix} 2 & 2  \
   9 & 4  \ \end{pmatrix} } \right)$$ and $$I = \left( {\begin{pmatrix} 1 & 0  \
   0 & 1  \ \end{pmatrix} } \right)$$ then 10A--1 is
equal to :

Accepted Answer

According to Cayley Hamilton equation

|A -- $$\lambda $$I| = 0

$$ \Rightarrow $$ $$\left| {\begin{pmatrix} {2 - \lambda } & 2  \
   9 & {4 - \lambda }  \

\end{pmatrix} } \right|$$ = 0

$$ \Rightarrow $$ (2 -- $$\lambda $$)(4 -- $$\lambda $$) -- 18 = 0

$$ \Rightarrow $$ 8 -- 2$$\lambda $$ -- 4$$\lambda $$ + $$\lambda $$2
 -- 18 = 0

$$ \Rightarrow $$ $$\lambda $$2
 -- 6$$\lambda $$ -- 10 = 0

$$ \therefore $$ A2
 -- 6A-- 10 = 0

$$ \Rightarrow $$ A--1(A2) -- 6A--1A -- 10A--1 = 0

$$ \Rightarrow $$ A -- 6I -- 10A--1 = 0

$$ \Rightarrow $$ 10A--1 = A -- 6I

Solutions · 1