Let A = `[(a,b),(c,d)] ,a,b,c,d in R`
If `A^(5)=A^(3)+I` then A is

Options

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

Answer

To solve the problem, we start with the given equation involving the matrix \( A \):

Given:
\[ A^5 = A^3 + I \]
where \( I \) is the identity matrix.

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate the identity matrix:
\[ A^5 - A^3 = I \]

### Step 2: Factoring the Left Side
Next, we can factor out \( A^3 \) from the left side:
\[ A^3(A^2 - I) = I \]

### Step 3: Analyzing the Equation
From this equation, we can deduce that \( A^3 \) must be invertible since it is multiplied by another matrix to yield the identity matrix. This implies that:
\[ A^3 \text{ is invertible} \]

### Step 4: Determining the Properties of A
Since \( A^3 \) is invertible, it follows that the determinant of \( A^3 \) is non-zero:
\[ \text{det}(A^3) \neq 0 \]
Using the property of determinants, we have:
\[ \text{det}(A^3) = (\text{det}(A))^3 \]
This means:
\[ (\text{det}(A))^3 \neq 0 \implies \text{det}(A) \neq 0 \]

### Step 5: Conclusion about A
Since the determinant of \( A \) is non-zero, we conclude that \( A \) is an invertible matrix.

### Final Answer
Thus, the matrix \( A \) is an **invertible matrix**.

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