7 Let A= [ajl22, where aj 0 for all i,j and A2 = 1. Let
a be the sum of all diagonal elements of A and b =
|A|. Then 3a? + 4b is equal to



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.

Let A = [a_ij] where a_ij ≥ 0 for all i,j and A^2 = I. Let a be the sum of all diagonal elements of A and b = |A|. Then 3a^2 + 4b is equal to?

Learn how to solve a problem involving a 2x2 involutory matrix, calculating the trace and determinant, and determining the value of 3a^2 + 4b. This solution covers matrix theory concepts such as the trace, determinant, and properties of involutory matrices.

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