If A is skew-symmetric matrix of order `2 and B=[(1,4),(2,9)] and c[(9,-4),(-2,1)]` respectively. Then `A^3 BC + A^5B^2C^2 + A^7B^3C^3 +.....+A^(2n+1) B^n C^n` where `n in N` is

Options

A. a symmetric matrix
B. a skew-symmetric matrix
C. an identity matrix
D. none of these

Explanation

To solve the problem, we need to evaluate the expression:

\[ S = A^3 BC + A^5 B^2 C^2 + A^7 B^3 C^3 + \ldots + A^{2n+1} B^n C^n \]

where \( A \) is a skew-symmetric matrix of order 2, and \( B \) and \( C \) are given matrices.

### Step 1: Understand the properties of skew-symmetric matrices

A skew-symmetric matrix \( A \) satisfies the property:

\[ A^T = -A \]

For a 2x2 skew-symmetric matrix, it can be represented as:

\[ A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix} \]

### Step 2: Compute \( BC \) and \( CB \)

Given:

\[ B = \begin{pmatrix} 1 & 4 \\ 2 & 9 \end{pmatrix}, \quad C = \begin{pmatrix} 9 & -4 \\ -2 & 1 \end{pmatrix} \]

First, we compute \( BC \):

\[ BC = \begin{pmatrix} 1 & 4 \\ 2 & 9 \end{pmatrix} \begin{pmatrix} 9 & -4 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 9 + 4 \cdot (-2) & 1 \cdot (-4) + 4 \cdot 1 \\ 2 \cdot 9 + 9 \cdot (-2) & 2 \cdot (-4) + 9 \cdot 1 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 9 - 8 = 1 \)
- First row, second column: \( -4 + 4 = 0 \)
- Second row, first column: \( 18 - 18 = 0 \)
- Second row, second column: \( -8 + 9 = 1 \)

Thus,

\[ BC = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \]

Now compute \( CB \):

\[ CB = \begin{pmatrix} 9 & -4 \\ -2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 4 \\ 2 & 9 \end{pmatrix} = \begin{pmatrix} 9 \cdot 1 + (-4) \cdot 2 & 9 \cdot 4 + (-4) \cdot 9 \\ -2 \cdot 1 + 1 \cdot 2 & -2 \cdot 4 + 1 \cdot 9 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 9 - 8 = 1 \)
- First row, second column: \( 36 - 36 = 0 \)
- Second row, first column: \( -2 + 2 = 0 \)
- Second row, second column: \( -8 + 9 = 1 \)

Thus,

\[ CB = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \]

### Step 3: Use the properties of \( BC \) and \( CB \)

Since \( BC = I \) and \( CB = I \), we can simplify the original expression \( S \):

\[ S = A^3 I + A^5 I + A^7 I + \ldots + A^{2n+1} I \]

This simplifies to:

\[ S = A^3 + A^5 + A^7 + \ldots + A^{2n+1} \]

### Step 4: Identify the pattern in powers of \( A \)

Since \( A \) is skew-symmetric, we know:

- \( A^2 \) is symmetric.
- \( A^3 \) is skew-symmetric.
- \( A^5 \) is skew-symmetric.
- \( A^{2n+1} \) is skew-symmetric.

### Step 5: Analyze the sum \( S \)

The sum \( S \) consists of skew-symmetric matrices:

\[ S = A^3 + A^5 + A^7 + \ldots + A^{2n+1} \]

### Step 6: Determine the nature of \( S \)

Since the sum of skew-symmetric matrices is also skew-symmetric, we conclude that:

\[ S \text{ is skew-symmetric.} \]

### Final Answer

Thus, the final answer is that \( S \) is a skew-symmetric matrix.