Let `M=[{:(0,1,a),(1,2,3),(3,b,1):}]`and adj `M =[{:(-1,,1,,-1),(8,,-6,,2),(-5,,3,,-1):}]`
where a and b are real numbers. Which of the following options is/are correct ?

Options

A. If `M[(alpha),(beta),(gamma)]=[(1),(2),(3)]` , then `alpha-beta+gamma=3`
B. `(adjM)^(-1) + adjM^(-1)=-M`
C. a+b=3
D. `det(adjM^2)=81`

Answer

To solve the problem, we need to analyze the given matrices \( M \) and \( \text{adj} M \) and find the values of \( a \) and \( b \). We will then check which of the given options are correct.

### Step 1: Write down the matrices
We have:
\[ M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix} \]
\[ \text{adj} M = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix} \]

### Step 2: Calculate the determinant of \( M \)
The determinant of a 3x3 matrix \( M \) can be calculated using the formula:
\[ \text{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix \( M \):
\[ \text{det}(M) = 0 \cdot (2 \cdot 1 - 3 \cdot b) - 1 \cdot (1 \cdot 1 - 3 \cdot 3) + a \cdot (1 \cdot b - 2 \cdot 3) \]
Calculating this gives:
\[ \text{det}(M) = 0 - (1 - 9) + a(b - 6) = 8 + a(b - 6) \]

### Step 3: Use the property of adjugate
We know that:
\[ M \cdot \text{adj} M = \text{det}(M) \cdot I \]
where \( I \) is the identity matrix. Thus, we can find the determinant of \( M \) by using the first element of the product \( M \cdot \text{adj} M \).

### Step 4: Calculate \( M \cdot \text{adj} M \)
Calculating the first element of \( M \cdot \text{adj} M \):
\[ M \cdot \text{adj} M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix} \cdot \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix} \]
Calculating the first row:
\[ 0 \cdot (-1) + 1 \cdot 8 + a \cdot (-5) = 8 - 5a \]
The first element of the result must equal \( \text{det}(M) \).

### Step 5: Set up equations
From the determinant calculation:
\[ 8 + a(b - 6) = 8 - 5a \]
This simplifies to:
\[ a(b - 6) + 5a = 0 \implies a(b - 6 + 5) = 0 \implies a(b - 1) = 0 \]
This gives us two cases:
1. \( a = 0 \)
2. \( b = 1 \)

### Step 6: Calculate the second equation using the second element
Using the second row of \( M \) and the second column of \( \text{adj} M \):
\[ 1 \cdot 1 + 2 \cdot (-6) + 3 \cdot 3 = 1 - 12 + 9 = -2 \]
This should equal \( \text{det}(M) \) as well.

### Step 7: Solve for \( a \) and \( b \)
From the calculations, we can conclude:
- If \( a = 0 \), then \( b \) can be any real number.
- If \( b = 1 \), then \( a \) can be any real number.

### Step 8: Check the options
Now we need to check the options given in the question:
1. \( a + b = 3 \)
2. \( a - b = 1 \)
3. \( a = 2 \)
4. \( b = 1 \)

From our findings:
- If \( a = 2 \), then \( b = 1 \) satisfies \( a + b = 3 \) and \( a - b = 1 \).
- Therefore, options 1, 2, and 3 are correct.

### Conclusion
The correct options are:
- \( a + b = 3 \) (True)
- \( a - b = 1 \) (True)
- \( a = 2 \) (True)
- \( b = 1 \) (True)