Search any question & find its solution
Question:
Answered & Verified by Expert
If matrix $D_1=\operatorname{diag}(a, b, c)$, matrix $D_2=\operatorname{diag}(3,3,3)$ and $\mathrm{A}$ is a skew symmetric matrix of $3^{\mathrm{rd}}$ order, then $\mathrm{Tr}$ $\left(D_1 D_2 A+D_1 D_2+D_1 A+D_2 A\right)-\operatorname{Tr}\left(D_1+D_2\right)=$
Options:
Solution:
2574 Upvotes
Verified Answer
The correct answer is:
$2 \mathrm{a}+2 \mathrm{~b}+2 \mathrm{c}-9$
$\because \mathrm{A}$ is skew symmetric matrix of $3^{\text {rd }}$ order
$\begin{aligned} & \because \operatorname{diag}(\mathrm{A})=(0,0,0) \\ & \operatorname{diag}\left(\mathrm{D}_1 \mathrm{D}_2 \mathrm{~A}+\mathrm{D}_1 \mathrm{D}_2+\mathrm{D}_1 \mathrm{~A}+\mathrm{D}_2 \mathrm{~A}\right)-\operatorname{dia}\left(\mathrm{D}_1+\mathrm{D}_2\right) \\ & =\operatorname{diag}(a \times 3 \times 0+a \times 3+a \times 0+3 \times 0, b \times 3 \times 0+b \times \\ & 3+b \times 0+3 \times 0, c \times 3 \times 0+c \times 3+c \times 0+3 \times 0) \\ & -\operatorname{diag}(3+a, b+3) \\ & =\operatorname{diag}(3 a, 3 b, 3 c)-\operatorname{diag}(3+a, b+3) \\ & =\operatorname{diag}(2 a-3,2 b-3,2 c-3) \\ & \text { Now, Tr }\left(\mathrm{D}_1 \mathrm{D}_2 \mathrm{~A}+\mathrm{D}_1 \mathrm{D}_2+\mathrm{D}_1 \mathrm{~A}+\mathrm{D}_2 \mathrm{~A}\right)-\operatorname{Tr}\left(\mathrm{D}_1+\mathrm{D}_2\right) \\ & =2 a-3+2 b-3+2 c-3=2 a+2 b+2 c-9\end{aligned}$
$\begin{aligned} & \because \operatorname{diag}(\mathrm{A})=(0,0,0) \\ & \operatorname{diag}\left(\mathrm{D}_1 \mathrm{D}_2 \mathrm{~A}+\mathrm{D}_1 \mathrm{D}_2+\mathrm{D}_1 \mathrm{~A}+\mathrm{D}_2 \mathrm{~A}\right)-\operatorname{dia}\left(\mathrm{D}_1+\mathrm{D}_2\right) \\ & =\operatorname{diag}(a \times 3 \times 0+a \times 3+a \times 0+3 \times 0, b \times 3 \times 0+b \times \\ & 3+b \times 0+3 \times 0, c \times 3 \times 0+c \times 3+c \times 0+3 \times 0) \\ & -\operatorname{diag}(3+a, b+3) \\ & =\operatorname{diag}(3 a, 3 b, 3 c)-\operatorname{diag}(3+a, b+3) \\ & =\operatorname{diag}(2 a-3,2 b-3,2 c-3) \\ & \text { Now, Tr }\left(\mathrm{D}_1 \mathrm{D}_2 \mathrm{~A}+\mathrm{D}_1 \mathrm{D}_2+\mathrm{D}_1 \mathrm{~A}+\mathrm{D}_2 \mathrm{~A}\right)-\operatorname{Tr}\left(\mathrm{D}_1+\mathrm{D}_2\right) \\ & =2 a-3+2 b-3+2 c-3=2 a+2 b+2 c-9\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.