D is a `3xx3` diagonal matrix. Which of the following
statements are not true?

Options

A. `D^(T)=D`
B. `AD=DA` for every matrix A of order `3xx3`
C. `D^(-1)` if exists is a scalar matrix
D. None of the above

Answer

B, C

Explanation

let, `D = [[a, 0, 0],[0,b,0],[0,0,c]] = D^(T) and "let " A= [[a_(1),a_(2) ,a_(3)],[b_(1) ,b_(2),b_(2) ],[c_(1) , c_(2) , c_(3)]]`
`therefore DA= [[a, 0, 0],[0,b,0],[0,0,c]] [[a_(1),a_(2) ,a_(3)],[b_(1) ,b_(2),b_(2) ],[c_(1) , c_(2) , c_(3)]] = [[aa_(1),aa_(2) ,aa_(3)],[b b_(1) ,b b_(2),b b_(2) ],[c c_(1) , c c_(2) , c c_(3)]]`
` AD= [[a_(1),a_(2) ,a_(3)],[b_(1) ,b_(2),b_(2) ],[c_(1) , c_(2) , c_(3)]][[a, 0, 0],[0,b,0],[0,0,c]] = [[a_(1)a,a_(2)b ,a_(3)c],[b_(1)a ,b_(2)b,b_(2)c ],[c_(1)a , c_(2)b , c_(3)c]] ne DA`
and `D^(-1) = [[1/a,0,0],[0,1/b,0],[0,0,1/c]]`
`abs(D^(-1)) = 1/(abc) ne 0 [becausea ne 0, b ne0, cne0]`