The number of `2xx2` matrices A with real entries, such that `A+A^(T)=3I` and `A A^(T)=5I`, is equal to

Options

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

Answer

To solve the problem, we need to find the number of \( 2 \times 2 \) matrices \( A \) with real entries such that:

1. \( A + A^T = 3I \)
2. \( A A^T = 5I \)

Let's denote the matrix \( A \) as:

\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

### Step 1: Solve the first equation \( A + A^T = 3I \)

The transpose of \( A \) is:

\[ A^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix} \]

Now, adding \( A \) and \( A^T \):

\[ A + A^T = \begin{pmatrix} a + a & b + c \\ c + b & d + d \end{pmatrix} = \begin{pmatrix} 2a & b + c \\ b + c & 2d \end{pmatrix} \]

Setting this equal to \( 3I \):

\[ \begin{pmatrix} 2a & b + c \\ b + c & 2d \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

From this, we can derive the following equations:

1. \( 2a = 3 \) → \( a = \frac{3}{2} \)
2. \( 2d = 3 \) → \( d = \frac{3}{2} \)
3. \( b + c = 0 \) → \( c = -b \)

### Step 2: Substitute into the second equation \( A A^T = 5I \)

Now, we need to compute \( A A^T \):

\[ A A^T = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & c \\ b & d \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & ac + bd \\ ca + db & c^2 + d^2 \end{pmatrix} \]

Substituting \( a = \frac{3}{2} \), \( d = \frac{3}{2} \), and \( c = -b \):

\[ A A^T = \begin{pmatrix} \left(\frac{3}{2}\right)^2 + b^2 & \frac{3}{2}(-b) + \frac{3}{2}b \\ \frac{3}{2}b + \frac{3}{2}(-b) & (-b)^2 + \left(\frac{3}{2}\right)^2 \end{pmatrix} \]

Calculating the elements:

1. \( \left(\frac{3}{2}\right)^2 + b^2 = \frac{9}{4} + b^2 \)
2. \( \frac{3}{2}(-b) + \frac{3}{2}b = 0 \)
3. \( \frac{3}{2}b + \frac{3}{2}(-b) = 0 \)
4. \( (-b)^2 + \left(\frac{3}{2}\right)^2 = b^2 + \frac{9}{4} \)

Thus, we have:

\[ A A^T = \begin{pmatrix} \frac{9}{4} + b^2 & 0 \\ 0 & b^2 + \frac{9}{4} \end{pmatrix} \]

Setting this equal to \( 5I \):

\[ \begin{pmatrix} \frac{9}{4} + b^2 & 0 \\ 0 & b^2 + \frac{9}{4} \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

From this, we get two equations:

1. \( \frac{9}{4} + b^2 = 5 \)
2. \( b^2 + \frac{9}{4} = 5 \)

Both equations are the same, so we can solve one:

\[ b^2 = 5 - \frac{9}{4} = \frac{20}{4} - \frac{9}{4} = \frac{11}{4} \]

Thus, \( b = \pm \frac{\sqrt{11}}{2} \).

### Step 3: Determine the matrices

Using \( b = \pm \frac{\sqrt{11}}{2} \) and \( c = -b \):

1. If \( b = \frac{\sqrt{11}}{2} \), then \( c = -\frac{\sqrt{11}}{2} \).
2. If \( b = -\frac{\sqrt{11}}{2} \), then \( c = \frac{\sqrt{11}}{2} \).

Thus, we have two matrices:

1. \( A_1 = \begin{pmatrix} \frac{3}{2} & \frac{\sqrt{11}}{2} \\ -\frac{\sqrt{11}}{2} & \frac{3}{2} \end{pmatrix} \)
2. \( A_2 = \begin{pmatrix} \frac{3}{2} & -\frac{\sqrt{11}}{2} \\ \frac{\sqrt{11}}{2} & \frac{3}{2} \end{pmatrix} \)

### Conclusion

The number of \( 2 \times 2 \) matrices \( A \) that satisfy the given conditions is **2**.