Let `A = [(0,2),(-2,0)]` and `(mI + nA)^2 = A` where m, n are positive real numbers and I is the identity matrix. What is `(m + n)` equal to?

Options

A. 0
B. `1/2`
C. `1`
D. `3/2`

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) such that \( (mI + nA)^2 = A \), where \( A = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} \) and \( I \) is the identity matrix.

### Step 1: Write down the identity matrix and the matrix \( A \)

The identity matrix \( I \) for a \( 2 \times 2 \) matrix is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
And the matrix \( A \) is given as:
\[ A = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} \]

### Step 2: Compute \( mI + nA \)

Now we compute \( mI + nA \):
\[ mI = \begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix} \]
\[ nA = n \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2n \\ -2n & 0 \end{pmatrix} \]
Adding these two matrices:
\[ mI + nA = \begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix} + \begin{pmatrix} 0 & 2n \\ -2n & 0 \end{pmatrix} = \begin{pmatrix} m & 2n \\ -2n & m \end{pmatrix} \]

### Step 3: Square the resulting matrix

Next, we need to square the matrix \( mI + nA \):
\[ (mI + nA)^2 = \begin{pmatrix} m & 2n \\ -2n & m \end{pmatrix} \begin{pmatrix} m & 2n \\ -2n & m \end{pmatrix} \]
Calculating the product:
- The element at (1,1):
\[ m \cdot m + 2n \cdot (-2n) = m^2 - 4n^2 \]
- The element at (1,2):
\[ m \cdot 2n + 2n \cdot m = 4mn \]
- The element at (2,1):
\[ -2n \cdot m + m \cdot (-2n) = -4mn \]
- The element at (2,2):
\[ -2n \cdot 2n + m \cdot m = -4n^2 + m^2 \]

Thus, we have:
\[ (mI + nA)^2 = \begin{pmatrix} m^2 - 4n^2 & 4mn \\ -4mn & m^2 - 4n^2 \end{pmatrix} \]

### Step 4: Set the squared matrix equal to \( A \)

Now we set this equal to matrix \( A \):
\[ \begin{pmatrix} m^2 - 4n^2 & 4mn \\ -4mn & m^2 - 4n^2 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} \]

### Step 5: Create equations from corresponding elements

From the equality of matrices, we can set up the following equations:
1. \( m^2 - 4n^2 = 0 \)
2. \( 4mn = 2 \)

### Step 6: Solve the equations

From the first equation:
\[ m^2 = 4n^2 \implies m = 2n \quad (\text{since } m, n > 0) \]

Substituting \( m = 2n \) into the second equation:
\[ 4(2n)n = 2 \implies 8n^2 = 2 \implies n^2 = \frac{1}{4} \implies n = \frac{1}{2} \]

Now substituting \( n \) back to find \( m \):
\[ m = 2n = 2 \cdot \frac{1}{2} = 1 \]

### Step 7: Calculate \( m + n \)

Finally, we find:
\[ m + n = 1 + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, \( m + n = \frac{3}{2} \).

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