Suppose that d is an n × n diagonal matrix with entries dii. show that d-1 is an n × n diagonal matrix with entries 1/dii.

For matrix D=[d11  0  0        0

                       0 d22 0         0

                       0   0   d33  0 0 0

                        .

                        .

                        0   0       0    0 0 . . . dnn]

Adj D =[d22d33..dnn 0  0 0 0

            0             allvaluesexceptd22 0 0 ...0

            0                  0                                   allvaluesexpectd33

            .

            .

                                                                                    allvaluesexceptdnn]

|D|=d11d22d33dnn

So D-1 = Adj D/|D|

So dividing we get:

D-1=[1/d11 0 0 0

       0       0 1/d22 ...0

      0       0    0   1/dnn]

Step-by-step explanation:

as we know that D-1=AdjD/|D|

and for adj more than 2cross2 we use cofactor method

to find cofactors of matrix we can get

leaving first row and first col

D11=d22d33d44...dnn (as its a diagonal matrix and determinant can be found by multiplying diagonal values)

D12=0 same as all upto D1n=0

D21=0 D22=d11d33d44...dnn all other D22 to D2n=0

and upto so on we can get cofactor matrix as :

[d22d33..dnn 0  0 0 0

            0             allvaluesexceptd22 0 0 ...0

            0                  0                                   allvaluesexpectd33

            .

            .

                                                                                    allvaluesexceptdnn]

Adj D is transpose of cofactor matrix which comes out to be the same :

[d22d33..dnn 0  0 0 0

            0             allvaluesexceptd22 0 0 ...0

            0                  0                                   allvaluesexpectd33

            .

            .

                                                                                    allvaluesexceptdnn]

Now comes determinant:

determinant of diagonal matrix can be calculated by multiplying all diagonal elements

So

|D|=d11d22d33d44...dnn

Solving for D-1 we get :

AdjD/|D|=[1/d11 0 0 0

       0       0 1/d22 ...0

      0       0    0   1/dnn]

i don't see any

step-by-step explanation:

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