If `A=[5a -b 3 2]` and `A (adj A)=A A^T` , then `5a+b` is equal to: (1) `-1` (2) 5 (3) 4 (4) 13

Options

A. -1
B. 5
C. 4
D. 13

Explanation

To solve the problem step by step, we will follow the outlined approach based on the information provided in the question and the video transcript.

### Step 1: Define the Matrix A
Given the matrix:
\[ A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix} \]

### Step 2: Find the Transpose of Matrix A
The transpose of matrix A, denoted as \( A^T \), is obtained by swapping rows with columns:
\[ A^T = \begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix} \]

### Step 3: Use the Property of Matrices
We know from the problem statement that:
\[ A \cdot (\text{adj } A) = A \cdot A^T \]
Using the property of determinants, we can express \( A \cdot (\text{adj } A) \) as:
\[ A \cdot (\text{adj } A) = \text{det}(A) \cdot I_2 \]
where \( I_2 \) is the identity matrix of order 2.

### Step 4: Calculate the Determinant of A
The determinant of A is calculated as follows:
\[ \text{det}(A) = (5a)(2) - (-b)(3) = 10a + 3b \]

### Step 5: Set Up the Equation
From the property we have:
\[ (10a + 3b) \cdot I_2 = A \cdot A^T \]
This means:
\[ \begin{bmatrix} 10a + 3b & 0 \\ 0 & 10a + 3b \end{bmatrix} = A \cdot A^T \]

### Step 6: Calculate \( A \cdot A^T \)
Now we compute \( A \cdot A^T \):
\[ A \cdot A^T = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix} \cdot \begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix} \]
Calculating the elements:
- First element: \( 5a \cdot 5a + (-b) \cdot (-b) = 25a^2 + b^2 \)
- Second element: \( 5a \cdot 3 + (-b) \cdot 2 = 15a - 2b \)
- Third element: \( 3 \cdot 5a + 2 \cdot (-b) = 15a - 2b \)
- Fourth element: \( 3 \cdot 3 + 2 \cdot 2 = 9 + 4 = 13 \)

Thus,
\[ A \cdot A^T = \begin{bmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{bmatrix} \]

### Step 7: Set the Two Matrices Equal
Now we equate the two matrices:
\[ \begin{bmatrix} 10a + 3b & 0 \\ 0 & 10a + 3b \end{bmatrix} = \begin{bmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{bmatrix} \]

From this, we can set up the following equations:
1. \( 10a + 3b = 25a^2 + b^2 \)
2. \( 10a + 3b = 13 \)
3. \( 15a - 2b = 0 \)

### Step 8: Solve the Equations
From the third equation, we can express \( b \) in terms of \( a \):
\[ b = \frac{15a}{2} \]

Substituting \( b \) into the second equation:
\[ 10a + 3\left(\frac{15a}{2}\right) = 13 \]
\[ 10a + \frac{45a}{2} = 13 \]
Multiplying through by 2 to eliminate the fraction:
\[ 20a + 45a = 26 \]
\[ 65a = 26 \implies a = \frac{26}{65} = \frac{2}{5} \]

Now substituting \( a \) back to find \( b \):
\[ b = \frac{15 \cdot \frac{2}{5}}{2} = 3 \]

### Step 9: Calculate \( 5a + b \)
Now we calculate \( 5a + b \):
\[ 5a + b = 5\left(\frac{2}{5}\right) + 3 = 2 + 3 = 5 \]

### Final Answer
Thus, the value of \( 5a + b \) is:
\[ \boxed{5} \]