A be a square matrix of order `2` with `|A| ne 0` such that `|A+|A|adj(A)|=0`, where `adj(A)` is a adjoint of matrix `A`, then the value of `|A-|A|adj(A)|` is

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

D

Explanation

`(d)` Let `A=[{:(m,n),(p,q):}]`,`adj(a)=[{:(q,-n),(-p,m):}]`
Let `|A|=d=mq-np`
`|A+dadjA|=|{:(m+qd,n(1-d)),(p(1-d),q+md):}|=0`
`impliesmq+m^(2)d+q^(2)d+mqd^(2)-np+2npd-npd^(2)=0`
`implies(mq-np)+(mq0np)d^(2)+m^(2)d+q^(2)d+2mqd-2d^(2)=0`
`implies(d+d^(3)-2d^(2))+d(m^(2)+q^(2)+2mq)=0`
Now, `|A- d adj|A|=-(m+q)^(2)+4(mq-np)=4d=4`