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Question

If D1 and D2 are two 3×3 diagonal matrices, then which of the following is/are true?

A
D1D2 is diagonal matrix
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B
D1D2=D2D1
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C
D21+D22 is a diagonal matrix
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D
none of the above
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Solution

The correct option is C D21+D22 is a diagonal matrix

Let D1=diag(a1,a2,a3) and
D2=diag(b1,b2,b3)

Now, D1D2=a1000a2000a3b1000b2000b3

D1D2=a1b1000a2b2000a3b3

D1D2=diag(a1b1,a2b2,a3b3)
which is a diagonal matrix.

D1D2=diag(b1a1,b2a2,b3a3)
=[diag(b1,b2,b3)][diag(a1,a2,a3)]
=D2D1
D1D2=D2D1

(D1)2=(diag(a1,a2,a3))2
=diag(a21,a22,a23)

(D2)2=diag(b1,b2,b3))2
=diag(b21,b22,b23)

D21+D22=diag(a21+b21,a22+b22,a23+b23)
which is also a diagonal matrix.

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