If D<sub>1</sub> and D<sub>2</sub> are two 3 × 3 diagonal matrices where none of the diagonal element is zero, then -
(A) D<sub>1</sub>D<sub>2</sub> is a diagonal matrix
(B) D<sub>1</sub>D<sub>2</sub> = D<sub>2</sub>D<sub>1</sub>
(C) D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> is a diagonal matrix
(D) D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> = (D<sub>1</sub> + D<sub>2</sub>)<sup>2</sup>
Step-by-Step Solution
Key Concept: Diagonal matrices commute with each other, and the product or sum of diagonal matrices is also a diagonal matrix.
Let D<sub>1</sub> = diag(a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>) and D<sub>2</sub> = diag(b<sub>1</sub>, b<sub>2</sub>, b<sub>3</sub>). Then D<sub>1</sub>D<sub>2</sub> = diag(a<sub>1</sub>b<sub>1</sub>, a<sub>2</sub>b<sub>2</sub>, a<sub>3</sub>b<sub>3</sub>), which is a diagonal matrix. Also, D<sub>2</sub>D<sub>1</sub> = diag(b<sub>1</sub>a<sub>1</sub>, b<sub>2</sub>a<sub>2</sub>, b<sub>3</sub>a<sub>3</sub>) = D<sub>1</sub>D<sub>2</sub>. D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> = diag(a<sub>1</sub><sup>2</sup>+b<sub>1</sub><sup>2</sup>, a<sub>2</sub><sup>2</sup>+b<sub>2</sub><sup>2</sup>, a<sub>3</sub><sup>2</sup>+b<sub>3</sub><sup>2</sup>), which is a diagonal matrix. (D<sub>1</sub>+D<sub>2</sub>)<sup>2</sup> = D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> + D<sub>1</sub>D<sub>2</sub> + D<sub>2</sub>D<sub>1</sub> = D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> + 2D<sub>1</sub>D<sub>2</sub>, which is not equal to D<sub>1</sub><sup>2</sup> + D<sub>2</sub><sup>2</sup> unless D<sub>1</sub>D<sub>2</sub> = 0, which is not true here.
Correct Answer: A,B,C