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 find the trace of the matrix expression \( C + C^3 + C^5 + \ldots + C^{99} \), we will follow these steps:

### Step 1: Define the Matrix C
The matrix \( C \) is given as:
\[ C = \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \]

### Step 2: Calculate \( C^2 \)
To find \( C^3 \), we first need to calculate \( C^2 \):
\[ C^2 = C \cdot C = \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \]

Calculating the elements of \( C^2 \):
- First row, first column: \( 1 \cdot 1 + 4 \cdot 7 + 6 \cdot 9 = 1 + 28 + 54 = 83 \)
- First row, second column: \( 1 \cdot 4 + 4 \cdot 2 + 6 \cdot 8 = 4 + 8 + 48 = 60 \)
- First row, third column: \( 1 \cdot 6 + 4 \cdot 5 + 6 \cdot 3 = 6 + 20 + 18 = 44 \)

- Second row, first column: \( 7 \cdot 1 + 2 \cdot 7 + 5 \cdot 9 = 7 + 14 + 45 = 66 \)
- Second row, second column: \( 7 \cdot 4 + 2 \cdot 2 + 5 \cdot 8 = 28 + 4 + 40 = 72 \)
- Second row, third column: \( 7 \cdot 6 + 2 \cdot 5 + 5 \cdot 3 = 42 + 10 + 15 = 67 \)

- Third row, first column: \( 9 \cdot 1 + 8 \cdot 7 + 3 \cdot 9 = 9 + 56 + 27 = 92 \)
- Third row, second column: \( 9 \cdot 4 + 8 \cdot 2 + 3 \cdot 8 = 36 + 16 + 24 = 76 \)
- Third row, third column: \( 9 \cdot 6 + 8 \cdot 5 + 3 \cdot 3 = 54 + 40 + 9 = 103 \)

Thus,
\[ C^2 = \begin{pmatrix} 83 & 60 & 44 \\ 66 & 72 & 67 \\ 92 & 76 & 103 \end{pmatrix} \]

### Step 3: Calculate \( C^3 \)
Next, we calculate \( C^3 = C^2 \cdot C \):
\[ C^3 = \begin{pmatrix} 83 & 60 & 44 \\ 66 & 72 & 67 \\ 92 & 76 & 103 \end{pmatrix} \cdot \begin{pmatrix} 1 & 4 & 6 \\ 7 & 2 & 5 \\ 9 & 8 & 3 \end{pmatrix} \]

Calculating the elements of \( C^3 \):
- First row, first column: \( 83 \cdot 1 + 60 \cdot 7 + 44 \cdot 9 = 83 + 420 + 396 = 899 \)
- First row, second column: \( 83 \cdot 4 + 60 \cdot 2 + 44 \cdot 8 = 332 + 120 + 352 = 804 \)
- First row, third column: \( 83 \cdot 6 + 60 \cdot 5 + 44 \cdot 3 = 498 + 300 + 132 = 930 \)

- Second row, first column: \( 66 \cdot 1 + 72 \cdot 7 + 67 \cdot 9 = 66 + 504 + 603 = 1173 \)
- Second row, second column: \( 66 \cdot 4 + 72 \cdot 2 + 67 \cdot 8 = 264 + 144 + 536 = 944 \)
- Second row, third column: \( 66 \cdot 6 + 72 \cdot 5 + 67 \cdot 3 = 396 + 360 + 201 = 957 \)

- Third row, first column: \( 92 \cdot 1 + 76 \cdot 7 + 103 \cdot 9 = 92 + 532 + 927 = 1551 \)
- Third row, second column: \( 92 \cdot 4 + 76 \cdot 2 + 103 \cdot 8 = 368 + 152 + 824 = 1344 \)
- Third row, third column: \( 92 \cdot 6 + 76 \cdot 5 + 103 \cdot 3 = 552 + 380 + 309 = 1241 \)

Thus,
\[ C^3 = \begin{pmatrix} 899 & 804 & 930 \\ 1173 & 944 & 957 \\ 1551 & 1344 & 1241 \end{pmatrix} \]

### Step 4: Calculate the Trace of \( C \)
The trace of a matrix is the sum of its diagonal elements. For matrix \( C \):
\[ \text{Trace}(C) = 1 + 2 + 3 = 6 \]

### Step 5: Calculate the Trace of \( C^3 \)
For matrix \( C^3 \):
\[ \text{Trace}(C^3) = 899 + 944 + 1241 = 3084 \]

### Step 6: Generalize for \( C^n \)
Notice that if \( C \) is a matrix with a trace of 0, then all odd powers of \( C \) will also have a trace of 0. Thus:
\[ \text{Trace}(C^1) = 6, \quad \text{Trace}(C^3) = 0, \quad \text{Trace}(C^5) = 0, \ldots, \text{Trace}(C^{99}) = 0 \]

### Step 7: Sum the Traces
Now we can sum the traces:
\[ \text{Trace}(C + C^3 + C^5 + \ldots + C^{99}) = \text{Trace}(C) + \text{Trace}(C^3) + \text{Trace}(C^5) + \ldots + \text{Trace}(C^{99}) = 6 + 0 + 0 + \ldots + 0 = 6 \]

### Final Answer
Thus, the trace of \( C + C^3 + C^5 + \ldots + C^{99} \) is:
\[ \boxed{6} \]