Let A be `a 3 xx 3` diagonal matrix which commutes
with ever `3xx3` matrix. If det (A) = 8 , then tr A is

Explanation

Let `A= [[alpha , 0, 0],[0, beta, 0],[0 ,0, gamma ]]`
`therefore [[alpha , 0, 0],[0, beta, 0],[0 ,0, gamma ]][[a,h,g],[h,b,f],[g,f,c]]=[[a,h,g],[h,b,f],[g,f,c]][[alpha , 0, 0],[0, beta, 0],[0 ,0, gamma ]]`
`rArr alpha = beta = gamma `
`therefore A=[[alpha , 0, 0],[0, alpha, 0],[0 ,0, alpha ]]`
`rArr` det `(A) = alpha^(3) = 8` [given]
`therefore alpha = 2`
`therefore A=[[2 , 0, 0],[0, 2, 0],[0 ,0, 2 ]]`
`rArr` tr A = 2 + 2+ 2 = 6