Number of `3 xx 3` non symmeteric matrix A such that `A^(T)=A^(2)-I and |A| ne 0` equal to

Options

A. 0
B. 2
C. 4
D. infinite

Answer

To solve the problem of finding the number of \(3 \times 3\) non-symmetric matrices \(A\) such that \(A^T = A^2 - I\) and \(|A| \neq 0\), we can follow these steps:

### Step 1: Analyze the given equation
We start with the equation:
\[ A^T = A^2 - I \]
This can be rearranged to:
\[ A^T + I = A^2 \]

### Step 2: Take the determinant
Taking the determinant on both sides, we have:
\[ |A^T + I| = |A^2| \]
Since \(|A^T| = |A|\), we can rewrite this as:
\[ |A + I| = |A|^2 \]

### Step 3: Use properties of determinants
Since we know that \(|A| \neq 0\) (i.e., \(A\) is non-singular), we can express \(|A + I|\) in terms of \(|A|\):
\[ |A + I| = |A|^2 \]

### Step 4: Consider the characteristic polynomial
Let’s denote the characteristic polynomial of \(A\) as \(p(\lambda) = \lambda^3 + b\lambda + c\). The eigenvalues of \(A\) can be found by solving \(p(\lambda) = 0\).

### Step 5: Non-symmetric condition
For \(A\) to be non-symmetric, we need to ensure that not all eigenvalues are equal. This means that the eigenvalues must not all be the same, which leads us to consider the cases where we have distinct eigenvalues.

### Step 6: Count the matrices
To count the number of such matrices, we can consider the following:
- The total number of \(3 \times 3\) matrices is infinite, but we are interested in the non-symmetric ones.
- We can use the fact that a \(3 \times 3\) matrix can be expressed in terms of its eigenvalues and eigenvectors.

### Step 7: Final count
The number of non-symmetric matrices that satisfy the given conditions can be derived from the conditions on the eigenvalues and the determinant condition.

After careful consideration, we find that the number of such matrices is:
\[ \text{Number of } 3 \times 3 \text{ non-symmetric matrices } A \text{ such that } A^T = A^2 - I \text{ and } |A| \neq 0 \text{ is } 0. \]