The element in the first row and third column of the inverse of the matrix `[(1,2,-3),(0,1,2),(0,0,1)]` is

Options

A. `-2`
B. 0
C. 1
D. 7

Explanation

To find the element in the first row and third column of 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 Matrix A

The determinant of a 3x3 matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is given by 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:

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

### Step 2: Calculate the Adjoint of Matrix A

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

1. **Cofactor of \(a_{11} = 1\)**:
\[ C_{11} = \text{det}\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = 1 \]

2. **Cofactor of \(a_{12} = 2\)**:
\[ C_{12} = -\text{det}\begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} = 0 \]

3. **Cofactor of \(a_{13} = -3\)**:
\[ C_{13} = \text{det}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = 0 \]

4. **Cofactor of \(a_{21} = 0\)**:
\[ C_{21} = -\text{det}\begin{pmatrix} 2 & -3 \\ 0 & 1 \end{pmatrix} = -2 \]

5. **Cofactor of \(a_{22} = 1\)**:
\[ C_{22} = \text{det}\begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix} = 1 \]

6. **Cofactor of \(a_{23} = 2\)**:
\[ C_{23} = -\text{det}\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} = 0 \]

7. **Cofactor of \(a_{31} = 0\)**:
\[ C_{31} = \text{det}\begin{pmatrix} 2 & -3 \\ 1 & 2 \end{pmatrix} = 4 \]

8. **Cofactor of \(a_{32} = 0\)**:
\[ C_{32} = -\text{det}\begin{pmatrix} 1 & -3 \\ 0 & 2 \end{pmatrix} = -2 \]

9. **Cofactor of \(a_{33} = 1\)**:
\[ C_{33} = \text{det}\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = 1 \]

Now, the cofactor matrix is:

\[ \text{Cof}(A) = \begin{pmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 4 & -2 & 1 \end{pmatrix} \]

Taking the transpose gives us the adjoint:

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

### Step 3: Calculate the Inverse of Matrix A

The inverse of matrix \(A\) is given by:

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

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

\[ A^{-1} = \text{adj}(A) = \begin{pmatrix} 1 & -2 & 4 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 4: Find the Element in the First Row and Third Column

The element in the first row and third column of \(A^{-1}\) is \(4\).

### Final Answer

The element in the first row and third column of the inverse of the matrix is \(4\).

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