If adj `[(1,0,2),(-1,1,-2),(0,2,1)]=[(5,a,-2),(1,1,0),(-2,-2,b)]` , then find a ,b

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the equation:

\[ \text{adj} \begin{pmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 5 & a & -2 \\ 1 & 1 & 0 \\ -2 & -2 & b \end{pmatrix} \]

### Step 1: Calculate the adjoint of the matrix

The adjoint of a matrix is the transpose of its cofactor matrix. We will calculate the cofactors for each element of the given matrix.

#### Finding Cofactors:

1. **Cofactor \( A_{11} \)**:
\[ A_{11} = \begin{vmatrix} 1 & -2 \\ 2 & 1 \end{vmatrix} = (1)(1) - (-2)(2) = 1 + 4 = 5 \]

2. **Cofactor \( A_{12} \)**:
\[ A_{12} = -\begin{vmatrix} -1 & -2 \\ 0 & 1 \end{vmatrix} = -((-1)(1) - (-2)(0)) = -(-1) = 1 \]

3. **Cofactor \( A_{13} \)**:
\[ A_{13} = \begin{vmatrix} -1 & 1 \\ 0 & 2 \end{vmatrix} = (-1)(2) - (1)(0) = -2 \]

4. **Cofactor \( A_{21} \)**:
\[ A_{21} = -\begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix} = -((0)(1) - (2)(2)) = -(-4) = 4 \]

5. **Cofactor \( A_{22} \)**:
\[ A_{22} = \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} = (1)(1) - (2)(0) = 1 \]

6. **Cofactor \( A_{23} \)**:
\[ A_{23} = -\begin{vmatrix} 1 & 0 \\ 0 & 2 \end{vmatrix} = -((1)(2) - (0)(0)) = -2 \]

7. **Cofactor \( A_{31} \)**:
\[ A_{31} = \begin{vmatrix} 0 & 2 \\ 1 & -2 \end{vmatrix} = (0)(-2) - (2)(1) = -2 \]

8. **Cofactor \( A_{32} \)**:
\[ A_{32} = -\begin{vmatrix} 1 & 2 \\ -1 & -2 \end{vmatrix} = -((1)(-2) - (2)(-1)) = -(-2 + 2) = 0 \]

9. **Cofactor \( A_{33} \)**:
\[ A_{33} = \begin{vmatrix} 1 & 0 \\ -1 & 1 \end{vmatrix} = (1)(1) - (0)(-1) = 1 \]

### Step 2: Form the cofactor matrix

The cofactor matrix is:

\[ \begin{pmatrix} 5 & 1 & -2 \\ 4 & 1 & -2 \\ -2 & 0 & 1 \end{pmatrix} \]

### Step 3: Transpose the cofactor matrix to get the adjoint

The adjoint matrix is the transpose of the cofactor matrix:

\[ \text{adj} = \begin{pmatrix} 5 & 4 & -2 \\ 1 & 1 & 0 \\ -2 & 0 & 1 \end{pmatrix} \]

### Step 4: Compare with the given adjoint matrix

Now we compare this adjoint matrix with the given matrix:

\[ \begin{pmatrix} 5 & a & -2 \\ 1 & 1 & 0 \\ -2 & -2 & b \end{pmatrix} \]

From the comparison, we can see:

1. \( a = 4 \)
2. \( b = 1 \)

### Final Answer

Thus, the values of \( a \) and \( b \) are:

\[ a = 4, \quad b = 1 \]