To solve this problem, let's break it down and analyze the given conditions step by step:

Given:

1. Matrix $A = [a_{ij}] \in \mathbb{R}^{2 \times 2}$, where $a_{ij} \geq 0$ for all $i, j$.
2. Matrix $A^2 = I$, where $I$ is the identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
3. $a$ is the sum of all diagonal elements of $A$, i.e., $a = \text{tr}(A)$, where $\text{tr}(A)$ denotes the trace of $A$.
4. $b = |A|$, where $|A|$ denotes the determinant of $A$.
5. We need to find the value of $3a^2 + 4b$.

Step-by-Step Solution:

1. Matrix Property: Since $A$ is a $2 \times 2$ matrix and satisfies $A^2 = I$, this means that $A$ is an involutory matrix (i.e., a matrix such that $A^2 = I$).

2. Structure of $A$: For a $2 \times 2$ matrix $A$, let’s assume: $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}.$ The condition $A^2 = I$ translates into: $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.$ Expanding this matrix multiplication yields two conditions:

   • $a_{11}^2 + a_{12}a_{21} = 1$
   • $a_{22}^2 + a_{12}a_{21} = 1$
   • $a_{11}a_{12} + a_{12}a_{22} = 0$
   • $a_{21}a_{11} + a_{21}a_{22} = 0$

3. Trace and Determinant:

   • The trace of $A$ (sum of diagonal elements): $a = a_{11} + a_{22}.$
   • The determinant of $A$: $b = \det(A) = a_{11}a_{22} - a_{12}a_{21}.$

4. Using the Relation $A^2 = I$:

   • From the condition $a_{11}^2 + a_{12}a_{21} = 1$ and $a_{22}^2 + a_{12}a_{21} = 1$, we get: $a_{11}^2 = a_{22}^2.$ This implies $a_{11} = a_{22}$ or $a_{11} = -a_{22}$.
   • If $a_{11} = a_{22}$, then $a = a_{11} + a_{11} = 2a_{11}$.
   • If $a_{11} = -a_{22}$, then $a = a_{11} - a_{11} = 0$.

5. Possible Values for $a$:

   • If $a_{11} = a_{22}$, $a = 2a_{11}$, and since $a_{12}a_{21} \geq 0$, $b = a_{11}^2 - a_{12}a_{21} = 1 - a_{12}a_{21} \geq 0$.
   • If $a = 0$, then $b = -a_{12}a_{21} = 1$.

6. Computing $3a^2 + 4b$:

   • If $a = 0$ and $b = 1$: $3a^2 + 4b = 3(0)^2 + 4(1) = 4.$
   • If $a = 2$ (for $a_{11} = a_{22} = 1$), then $b = 0$: $3a^2 + 4b = 3(2)^2 + 4(0) = 3 \times 4 = 12.$

Since the problem states to find $3a^2 + 4b$, the possible values are 4 and 12 depending on the structure of $A$. Usually, without additional restrictions, we consider both cases.

Would you like more details on the derivation, or do you have any further questions?

Related Questions:

1. What is an involutory matrix, and why is it important in this problem?
2. How do you compute the trace and determinant of a $2 \times 2$ matrix?
3. What other types of matrices satisfy the property $A^2 = I$?
4. How does the trace of a matrix relate to its eigenvalues?
5. What are the conditions for a matrix to have non-negative entries?

Tip:

When dealing with matrices that satisfy specific properties (like $A^2 = I$), always try to explore potential forms or values for diagonal and off-diagonal elements to simplify the problem.