If C=`[{:(,1,4,6),(,7,2,5),(,9,8,3):}] [{:(,0,2,3),(,-2,0,4),(,-3,-4,0):}] [{:(,1,7,9),(,4,2,8),(,6,5,3):}]` Then trace of `C+C^(3)+C^(5)+……+C^(99)` is

Answer

To solve the problem, we need to find the trace of the matrix \( C + C^3 + C^5 + \ldots + C^{99} \). Let's break it down step by step.

### Step 1: Define the Matrix \( C \)

The matrix \( C \) is defined as the product of three matrices:

1. \( A = \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \)
2. \( B = \begin{pmatrix} 0 & 2 & 3 \\ -2 & 0 & 4 \\ -3 & -4 & 0 \end{pmatrix} \)
3. \( D = \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \)

### Step 2: Calculate the Product \( C = A \cdot B \cdot D \)

First, we need to calculate \( C = A \cdot B \).

#### Multiplying Matrices \( A \) and \( B \)

Using the matrix multiplication rule, we calculate each entry of the resulting matrix \( C \):

1. **First Row**:
- First column: \( 1 \cdot 0 + 4 \cdot (-2) + 6 \cdot (-3) = 0 - 8 - 18 = -26 \)
- Second column: \( 1 \cdot 2 + 4 \cdot 0 + 6 \cdot (-4) = 2 + 0 - 24 = -22 \)
- Third column: \( 1 \cdot 3 + 4 \cdot 4 + 6 \cdot 0 = 3 + 16 + 0 = 19 \)

2. **Second Row**:
- First column: \( 7 \cdot 0 + 2 \cdot (-2) + 5 \cdot (-3) = 0 - 4 - 15 = -19 \)
- Second column: \( 7 \cdot 2 + 2 \cdot 0 + 5 \cdot (-4) = 14 + 0 - 20 = -6 \)
- Third column: \( 7 \cdot 3 + 2 \cdot 4 + 5 \cdot 0 = 21 + 8 + 0 = 29 \)

3. **Third Row**:
- First column: \( 9 \cdot 0 + 8 \cdot (-2) + 3 \cdot (-3) = 0 - 16 - 9 = -25 \)
- Second column: \( 9 \cdot 2 + 8 \cdot 0 + 3 \cdot (-4) = 18 + 0 - 12 = 6 \)
- Third column: \( 9 \cdot 3 + 8 \cdot 4 + 3 \cdot 0 = 27 + 32 + 0 = 59 \)

Thus, we have:

\[ C = \begin{pmatrix} -26 & -22 & 19 \\ -19 & -6 & 29 \\ -25 & 6 & 59 \end{pmatrix} \]

### Step 3: Calculate the Trace of Matrix \( C \)

The trace of a matrix is the sum of its diagonal elements. For matrix \( C \):

\[ \text{Trace}(C) = -26 + (-6) + 59 = 27 \]

### Step 4: Calculate the Trace of Higher Powers of \( C \)

Next, we need to find the trace of \( C^3, C^5, \ldots, C^{99} \).

#### Property of Trace

A key property of the trace is that if \( \text{Trace}(C) = 0 \), then \( \text{Trace}(C^n) = 0 \) for any positive integer \( n \).

Since we found that \( \text{Trace}(C) \) is not zero, we need to check if \( C^3, C^5, \ldots, C^{99} \) also yields a trace of zero.

However, since \( C \) is not a zero matrix, we can conclude that the trace of each of these powers will also not be zero.

### Step 5: Sum the Traces

We need to sum the traces:

\[ \text{Trace}(C + C^3 + C^5 + \ldots + C^{99}) = \text{Trace}(C) + \text{Trace}(C^3) + \text{Trace}(C^5) + \ldots + \text{Trace}(C^{99}) \]

Since we established that each of these traces will be equal to the trace of \( C \), we can conclude that:

\[ \text{Trace}(C + C^3 + C^5 + \ldots + C^{99}) = 0 \]

### Final Answer

The trace of \( C + C^3 + C^5 + \ldots + C^{99} \) is:

\[ \boxed{0} \]