Find the characteristic equation of the matrix` A= [(2,1),(3,2)]` and hence find its inverse using Cayley-hamilton theorem.

Explanation

Charaacteritic equation is
`|A-lambdaI|=0rArr [(2-lambda,1),(3,2-lambda)]=0`
`rArr (2-lambda)^(2)-3=0`
`rArr lambda^(2)-4lambda+1=0`
therefore Cayley-hamiltion theorem,
`A^(2)-4A+I=O or I=4A-A^(2)`
Multiplying by` A^(-1)`, we get
` A^(-1)=4A^(-1)A-A^(-1)A A`
=` 4I-IA=4I-A`
=`4[(1,0),(0,1)]-{(2,1),(3,2)]`
` [(2,-1),(-3,2)]`