If `A=[(1,2,3),(0,1,2),(0,0,1)]` then `A^(-1)`=………..

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of \( A \)

The determinant of a \( 3 \times 3 \) matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

\[ \text{det}(A) = 1 \cdot (1 \cdot 1 - 2 \cdot 0) - 2 \cdot (0 \cdot 1 - 2 \cdot 0) + 3 \cdot (0 \cdot 0 - 1 \cdot 0) \]

Calculating this gives:

\[ \text{det}(A) = 1 \cdot (1 - 0) - 2 \cdot (0 - 0) + 3 \cdot (0 - 0) = 1 \]

### Step 2: Calculate the Cofactor Matrix of \( A \)

The cofactor \( C_{ij} \) is calculated using the formula:

\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]

where \( M_{ij} \) is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column.

Calculating the cofactors:

- \( C_{11} = \text{det} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = 1 \)
- \( C_{12} = -\text{det} \begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} = 0 \)
- \( C_{13} = \text{det} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = 0 \)

- \( C_{21} = -\text{det} \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix} = -2 \)
- \( C_{22} = \text{det} \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix} = 1 \)
- \( C_{23} = -\text{det} \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} = 0 \)

- \( C_{31} = \text{det} \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} = 1 \)
- \( C_{32} = -\text{det} \begin{pmatrix} 1 & 3 \\ 0 & 2 \end{pmatrix} = -2 \)
- \( C_{33} = \text{det} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = 1 \)

Thus, the cofactor matrix \( C \) is:

\[ C = \begin{pmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{pmatrix} \]

### Step 3: Calculate the Adjoint of \( A \)

The adjoint of \( A \) is the transpose of the cofactor matrix:

\[ \text{adj}(A) = C^T = \begin{pmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 4: Calculate the Inverse of \( A \)

Finally, we use the formula for the inverse of a matrix:

\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]

Since \( \text{det}(A) = 1 \):

\[ A^{-1} = 1 \cdot \begin{pmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix} \]

### Final Answer

\[ A^{-1} = \begin{pmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix} \]