If `A=[(3,-3,4),(2,-3,4),(0,-1,1)]`, then `A^(-1)=`

Options

A. A
B. `A^(2)`
C. `A^(3)`
D. `A^(4)`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix} \), we will follow these steps:

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

\[ \text{det}(A) = 3((-3)(1) - (4)(-1)) - (-3)((2)(1) - (4)(0)) + 4((2)(-1) - (-3)(0)) \]

Calculating each term:

1. \( 3((-3)(1) - (4)(-1)) = 3(-3 + 4) = 3(1) = 3 \)
2. \( -(-3)((2)(1) - (4)(0)) = 3(2) = 6 \)
3. \( 4((2)(-1) - (-3)(0)) = 4(-2) = -8 \)

Now, summing these up:

\[ \text{det}(A) = 3 + 6 - 8 = 1 \]

### Step 2: Find the Adjugate of Matrix A

The adjugate of a matrix is the transpose of the cofactor matrix. We will find the cofactor for each element of matrix \( A \).

1. **Cofactor of \( a_{11} = 3 \)**:
\[ C_{11} = \text{det}\begin{pmatrix} -3 & 4 \\ -1 & 1 \end{pmatrix} = (-3)(1) - (4)(-1) = -3 + 4 = 1 \]

2. **Cofactor of \( a_{12} = -3 \)**:
\[ C_{12} = -\text{det}\begin{pmatrix} 2 & 4 \\ 0 & 1 \end{pmatrix} = -((2)(1) - (4)(0)) = -2 \]

3. **Cofactor of \( a_{13} = 4 \)**:
\[ C_{13} = \text{det}\begin{pmatrix} 2 & -3 \\ 0 & -1 \end{pmatrix} = (2)(-1) - (-3)(0) = -2 \]

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

5. **Cofactor of \( a_{22} = -3 \)**:
\[ C_{22} = \text{det}\begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix} = (3)(1) - (4)(0) = 3 \]

6. **Cofactor of \( a_{23} = 4 \)**:
\[ C_{23} = -\text{det}\begin{pmatrix} 3 & -3 \\ 0 & -1 \end{pmatrix} = -((3)(-1) - (-3)(0)) = 3 \]

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

8. **Cofactor of \( a_{32} = -1 \)**:
\[ C_{32} = -\text{det}\begin{pmatrix} 3 & 4 \\ 2 & 4 \end{pmatrix} = -((3)(4) - (4)(2)) = 0 \]

9. **Cofactor of \( a_{33} = 1 \)**:
\[ C_{33} = \text{det}\begin{pmatrix} 3 & -3 \\ 2 & -3 \end{pmatrix} = (3)(-3) - (-3)(2) = -9 + 6 = -3 \]

Now, the cofactor matrix is:

\[ C = \begin{pmatrix} 1 & -2 & -2 \\ -1 & 3 & 3 \\ 0 & 0 & -3 \end{pmatrix} \]

Taking the transpose gives us the adjugate:

\[ \text{adj}(A) = \begin{pmatrix} 1 & -1 & 0 \\ -2 & 3 & 0 \\ -2 & 3 & -3 \end{pmatrix} \]

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

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

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

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

\[ A^{-1} = \text{adj}(A) = \begin{pmatrix} 1 & -1 & 0 \\ -2 & 3 & 0 \\ -2 & 3 & -3 \end{pmatrix} \]

### Final Answer

Thus, the inverse of matrix \( A \) is:

\[ A^{-1} = \begin{pmatrix} 1 & -1 & 0 \\ -2 & 3 & 0 \\ -2 & 3 & -3 \end{pmatrix} \]