1. Key Concepts and Formulas

• Involutory Matrix: A square matrix $A$ is called involutory if $A^2 = I$, where $I$ is the identity matrix of the same order.
• Properties of Determinants: For any square matrices $M$ and $N$ of the same order, $|MN| = |M||N|$. Consequently, for any positive integer $k$, $|M^k| = (|M|)^k$. Also, the determinant of an identity matrix is always 1, i.e., $|I|=1$.
• Trace of a Matrix: The trace of a square matrix $A$, denoted as $\mathrm{Tr}(A)$, is the sum of its diagonal elements. For a $2 \times 2$ matrix $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, $\mathrm{Tr}(A) = a_{11} + a_{22}$.

2. Step-by-Step Solution

Step 1: Understand the given matrix and conditions. We are given a $2 \times 2$ matrix $A$: $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ The problem states that $a_{ij} \neq 0$ for all $i, j$. This is a crucial condition. We are also given that $A^2 = I$, where $I$ is the $2 \times 2$ identity matrix: $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ We need to find the value of $3a^2 + 4b^2$, where $a$ is the sum of the diagonal elements of $A$ (trace of $A$), and $b$ is the determinant of $A$. So, $a = a_{11} + a_{22}$ and $b = |A| = a_{11}a_{22} - a_{12}a_{21}$.

Step 2: Calculate $A^2$ and equate it to the identity matrix. First, let's compute $A^2$: $A^2 = A \cdot A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ $A^2 = \begin{bmatrix} a_{11}a_{11} + a_{12}a_{21} & a_{11}a_{12} + a_{12}a_{22} \\ a_{21}a_{11} + a_{22}a_{21} & a_{21}a_{12} + a_{22}a_{22} \end{bmatrix}$ $A^2 = \begin{bmatrix} a_{11}^2 + a_{12}a_{21} & a_{12}(a_{11} + a_{22}) \\ a_{21}(a_{11} + a_{22}) & a_{21}a_{12} + a_{22}^2 \end{bmatrix}$ Now, we equate this to the identity matrix $I$: $\begin{bmatrix} a_{11}^2 + a_{12}a_{21} & a_{12}(a_{11} + a_{22}) \\ a_{21}(a_{11} + a_{22}) & a_{21}a_{12} + a_{22}^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ This equality gives us a system of four equations:

1. $a_{11}^2 + a_{12}a_{21} = 1$
2. $a_{12}(a_{11} + a_{22}) = 0$
3. $a_{21}(a_{11} + a_{22}) = 0$
4. $a_{21}a_{12} + a_{22}^2 = 1$

Step 3: Determine the value of 'a' (sum of diagonal elements). Recall that $a = a_{11} + a_{22}$. We can find the value of $a$ by examining equations (2) and (3). From equation (2): $a_{12}(a_{11} + a_{22}) = 0$. The problem statement explicitly says that $a_{ij} \neq 0$ for all $i,j$. Therefore, $a_{12} \neq 0$. For the product of two non-zero numbers to be zero, one of them must be zero. Since $a_{12} \neq 0$, it implies that the other factor must be zero: $a_{11} + a_{22} = 0$ Similarly, from equation (3): $a_{21}(a_{11} + a_{22}) = 0$. Since $a_{21} \neq 0$, this also leads to: $a_{11} + a_{22} = 0$ Both equations consistently show that $a_{11} + a_{22} = 0$. Since $a$ is defined as $a_{11} + a_{22}$, we have found: $a = 0$

Step 4: Determine the value of 'b' (determinant of A). We are given $A^2 = I$. To find $b = |A|$, we can take the determinant of both sides of this equation. $|A^2| = |I|$ Using the property that $|M^k| = (|M|)^k$, we can write $|A^2|$ as $(|A|)^2$. Also, we know that $|I|=1$ for any identity matrix. $(|A|)^2 = 1$ Since $b = |A|$, we can substitute this into the equation: $b^2 = 1$ This implies $b = 1$ or $b = -1$. For the expression $3a^2 + 4b^2$, we only need the value of $b^2$, which is $1$.

Step 5: Calculate the final expression $3a^2 + 4b^2$. Now we have the values:

• $a = 0$
• $b^2 = 1$ Substitute these values into the expression $3a^2 + 4b^2$: $3a^2 + 4b^2 = 3(0)^2 + 4(1)$ $= 3(0) + 4$ $= 0 + 4$ $= 4$

3. Common Mistakes & Tips

• Ignoring $a_{ij} \neq 0$: This condition is vital. If $a_{12}$ or $a_{21}$ could be zero, then $a_{11}+a_{22}$ would not necessarily be zero. For example, if $A=I$, then $a_{11}+a_{22}=2$, but $a_{12}=a_{21}=0$, which is ruled out here.
• Properties of Involutory Matrices: For a $2 \times 2$ matrix $A$ such that $A^2=I$ and all its elements are non-zero, it is a special case. Its trace (sum of diagonal elements) is always 0, and its determinant is always -1. Recognizing this can save time.
• Cayley-Hamilton Theorem (Advanced Insight): For a $2 \times 2$ matrix $A$, the characteristic equation is $\lambda^2 - \mathrm{Tr}(A)\lambda + |A| = 0$. By the Cayley-Hamilton Theorem, $A^2 - \mathrm{Tr}(A)A + |A|I = 0$. Given $A^2=I$, we substitute to get $I - \mathrm{Tr}(A)A + |A|I = 0$, which rearranges to $(1+|A|)I = \mathrm{Tr}(A)A$. Since $A$ has non-zero off-diagonal elements, it cannot be a scalar multiple of $I$ (unless $A=\pm I$, which would imply zero off-diagonal elements). Thus, $\mathrm{Tr}(A)$ must be $0$. If $\mathrm{Tr}(A)=0$, then $(1+|A|)I = 0$, which means $1+|A|=0 \implies |A|=-1$. This confirms $a=0$ and $b=-1$ (so $b^2=1$).

4. Summary

By leveraging the condition $A^2=I$ and the critical information that all elements $a_{ij}$ are non-zero, we systematically found the sum of the diagonal elements ($a$) and the determinant ($b$). Calculating $A^2$ and equating it to the identity matrix allowed us to deduce that $a = a_{11} + a_{22} = 0$. Taking the determinant of $A^2=I$ revealed that $b^2 = (|A|)^2 = 1$. Finally, substituting these values into the expression $3a^2+4b^2$ yielded the result $4$.

The final answer is $\boxed{4}$, which corresponds to option (A).