Question:
If matrix \( D_1 = \text{diag}(a, b, c) \), matrix \( D_2 = \text{diag}(3, 3, 3) \) and \( A \) is a skew-symmetric matrix of order 3, then
\[
\text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2) = ?
\]
If matrix \( D_1 = \text{diag}(a, b, c) \), matrix \( D_2 = \text{diag}(3, 3, 3) \) and \( A \) is a skew-symmetric matrix of order 3, then
\[ \text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2) = ? \]
\[ \text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2) = ? \]
Show Hint
When dealing with traces of matrix products, remember that the trace of a sum is the sum of the traces, and use properties of skew-symmetric matrices to simplify the problem.
Updated On: May 13, 2025
- \( 2a + 2b + 2c - 9 \)
- \( 3a + 3b + 3c - 9 \)
- \( 3a + 3b + 3c \)
- \( a^3 + b^3 + c^3 \)
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The Correct Option is A
Solution and Explanation
We are given the matrices \( D_1 = \text{diag}(a, b, c) \), \( D_2 = \text{diag}(3, 3, 3) \), and \( A \) as a skew-symmetric matrix of order 3.
A skew-symmetric matrix \( A \) satisfies \( A^T = -A \), which implies that the diagonal elements of \( A \) are all zero.
We need to evaluate the expression:
\[
\text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2)
\]
Using properties of the trace and the fact that \( A \) is skew-symmetric, we know:
\[
\text{Tr}(D_1 D_2 A) = 0, \quad \text{Tr}(D_2 D_1) = 0, \quad \text{Tr}(D_1 A) = 0, \quad \text{Tr}(D_2 A) = 0
\]
Thus, the expression simplifies to:
\[
\text{Tr}(D_1 + D_2) = \text{Tr}\left( \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} + \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \right)
\]
\[
= \text{Tr}\left( \begin{bmatrix} a+3 & 0 & 0 \\ 0 & b+3 & 0 \\ 0 & 0 & c+3 \end{bmatrix} \right)
\]
\[
= (a+3) + (b+3) + (c+3) = a + b + c + 9
\]
Thus, the expression becomes:
\[
0 - (a + b + c + 9) = - (a + b + c + 9) = 2a + 2b + 2c - 9
\]
Therefore, the correct answer is \( 2a + 2b + 2c - 9 \).
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