The number of matrices
`A = [(a,b),(c,d)]` ( where a,b,c,d`in` R ) such that `A^(-1)` = -A is :

Options

A. 0
B. 1
C. 2
D. infinite

Answer

To solve the problem of finding the number of matrices \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) such that \( A^{-1} = -A \), we can follow these steps:

### Step 1: Use the property of the inverse
Given that \( A^{-1} = -A \), we can multiply both sides by \( A \):
\[ A A^{-1} = A (-A) \]
This simplifies to:
\[ I = -A^2 \]
where \( I \) is the identity matrix.

### Step 2: Rearranging the equation
From the previous step, we can rearrange the equation:
\[ A^2 = -I \]
This means that the square of matrix \( A \) is equal to the negative of the identity matrix.

### Step 3: Express \( A^2 \)
Let’s express \( A^2 \) in terms of its elements:
\[ A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix} \]
Setting this equal to \( -I \) gives us:
\[ \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]

### Step 4: Set up the equations
From the equality of the matrices, we can derive the following equations:
1. \( a^2 + bc = -1 \)
2. \( ab + bd = 0 \)
3. \( ac + cd = 0 \)
4. \( bc + d^2 = -1 \)

### Step 5: Analyze the equations
From the second equation \( ab + bd = 0 \), we can factor it:
\[ b(a + d) = 0 \]
This gives us two cases:
- Case 1: \( b = 0 \)
- Case 2: \( a + d = 0 \) (i.e., \( d = -a \))

From the third equation \( ac + cd = 0 \), we can also factor it:
\[ c(a + d) = 0 \]
This gives us two cases:
- Case 1: \( c = 0 \)
- Case 2: \( a + d = 0 \) (i.e., \( d = -a \))

### Step 6: Consider the implications
- If \( b = 0 \) and \( c = 0 \), then the first and fourth equations become:
- \( a^2 = -1 \) (not possible for real \( a \))
- \( d^2 = -1 \) (not possible for real \( d \))

- If \( a + d = 0 \) (i.e., \( d = -a \)), we can substitute \( d \) in the equations:
- From \( a^2 + bc = -1 \)
- From \( bc + d^2 = -1 \) becomes \( bc + a^2 = -1 \)

### Step 7: Conclude the solution
Both equations \( a^2 + bc = -1 \) and \( bc + a^2 = -1 \) are satisfied for any real numbers \( a \) and \( b \) (with \( c \) being dependent on \( b \)). Therefore, there are infinitely many matrices \( A \) that satisfy the condition \( A^{-1} = -A \).

### Final Answer
The number of matrices \( A \) such that \( A^{-1} = -A \) is **infinite**.

---