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.