For an involutory matrix A (elements of whose principal diagonal are not same) of order 2 × 2, the value of 3(trace (A))² + 4(det (A))² is equal to

Let matrix \( A \) be an involutory matrix, meaning \( A^2 = I \), where \( I \) is the identity matrix. For a \( 2 \times 2 \) matrix, we will calculate the value of \( 3(\text{trace}(A))^2 + 4(\text{det}(A))^2 \).

### Step 1: General form of a \( 2 \times 2 \) matrix
Let \( A \) be a general \( 2 \times 2 \) matrix:

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

For an involutory matrix, we know \( A^2 = I \). Let's use this property.

### Step 2: Condition for an involutory matrix
From \( A^2 = I \), we expand \( A^2 \):

\[
A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix}
\]

Equating this to the identity matrix \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), we get the following system of equations:

1. \( a^2 + bc = 1 \)
2. \( ab + bd = 0 \)
3. \( ac + cd = 0 \)
4. \( bc + d^2 = 1 \)

### Step 3: Trace and determinant of \( A \)
The trace of matrix \( A \) is:

\[
\text{trace}(A) = a + d
\]

The determinant of matrix \( A \) is:

\[
\text{det}(A) = ad - bc
\]

### Step 4: Squaring trace and determinant
We are asked to find the value of:

\[
3(\text{trace}(A))^2 + 4(\text{det}(A))^2
\]

Let \( t = \text{trace}(A) = a + d \) and \( \Delta = \text{det}(A) = ad - bc \). Then the expression becomes:

\[
3t^2 + 4\Delta^2
\]

### Step 5: Using the involutory property
From the property \( A^2 = I \), it can be shown that for a \( 2 \times 2 \) involutory matrix, the trace \( t \) satisfies:

\[
t^2 = 4
\]

And the determinant of an involutory matrix is always:

\[
\Delta = 1
\]

### Step 6: Substituting values
Now substitute \( t^2 = 4 \) and \( \Delta = 1 \) into the expression:

\[
3(4) + 4(1)^2 = 12 + 4 = 16
\]

### Final Answer:
The value of \( 3(\text{trace}(A))^2 + 4(\text{det}(A))^2 \) is \( \boxed{16} \).

Would you like more details or have any questions about the solution?

Here are 5 related questions to expand your understanding:
1. What is an involutory matrix and how can we recognize it?
2. Why is the determinant of an involutory matrix always 1?
3. Can a non-diagonal matrix be involutory? Give an example.
4. How does the trace of a matrix relate to its eigenvalues?
5. Can we generalize this result to higher-dimensional involutory matrices?

**Tip:** When solving matrix equations, using the matrix's properties (like trace, determinant, and special conditions like involutory) can significantly simplify the computation.

Explore the solution for finding the value of 3(trace(A))² + 4(det(A))² for a 2x2 involutory matrix. This problem involves matrix algebra, the properties of involutory matrices, and the calculation of trace and determinant. Ideal for undergraduate-level students.

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