The inverse of $$ \left[\begin{array}{ccc}-1 & -2 & 5 \ -4 & 9 & -18 \ 1 & -2 & 4\end{array} ight] $$ is

Question

The inverse of $$ \left[\begin{array}{ccc}-1 & -2 & 5 \ -4 & 9 & -18 \ 1 & -2 & 4\end{array}ight] $$ is

UpStudy AI · Accepted Answer

To find the inverse of the matrix

$$
A = \begin{bmatrix}
-1 & -2 & 5 \
-4 & 9 & -18 \
1 & -2 & 4
\end{bmatrix},
$$

we can use the formula for the inverse of a $$3 \times 3$$ matrix, which involves calculating the matrix of cofactors, taking its transpose (the adjugate), and then dividing by the determinant of $$A$$.

### Step 1: Calculate the Determinant of $$A$$

The determinant of $$A$$ is calculated as follows:

$$
\text{det}(A) = -1 \cdot \begin{vmatrix} 9 & -18 \ -2 & 4 \end{vmatrix} - (-2) \cdot \begin{vmatrix} -4 & -18 \ 1 & 4 \end{vmatrix} + 5 \cdot \begin{vmatrix} -4 & 9 \ 1 & -2 \end{vmatrix}
$$

$$
= -1 \cdot (9 \times 4 - (-18) \times (-2)) + 2 \cdot ((-4 \times 4) - (-18) \times 1) + 5 \cdot ((-4 \times -2) - 9 \times 1)
$$

$$
= -1 \cdot (36 - 36) + 2 \cdot (-16 + 18) + 5 \cdot (8 - 9)
$$

$$
= 0 + 4 - 5 = -1
$$

Since $$\text{det}(A) \neq 0$$, the inverse of $$A$$ exists.

### Step 2: Calculate the Matrix of Cofactors

The matrix of cofactors $$C$$ is:

$$
C = \begin{bmatrix}
0 & -2 & -1 \
-2 & -9 & -4 \
-9 & -38 & -17
\end{bmatrix}
$$

### Step 3: Transpose the Matrix of Cofactors (Adjugate)

The adjugate of $$A$$ is the transpose of $$C$$:

$$
\text{adj}(A) = C^T = \begin{bmatrix}
0 & -2 & -9 \
-2 & -9 & -38 \
-1 & -4 & -17
\end{bmatrix}
$$

### Step 4: Calculate the Inverse of $$A$$

Finally, the inverse of $$A$$ is:

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

$$
A^{-1} = \begin{bmatrix}
0 & 2 & 9 \
2 & 9 & 38 \
1 & 4 & 17
\end{bmatrix}
$$

### Conclusion

The inverse of the given matrix $$A$$ is:

$$
A^{-1} = \begin{bmatrix}
\,0\; &\, 2\; &\, 9\, \
\,2\; &\, 9\; &\, 38\, \
\,1\; &\, 4\; &\, 17\,
\end{bmatrix}
$$
The inverse of the matrix is: $$ \begin{bmatrix} 0 & 2 & 9 \ 2 & 9 & 38 \ 1 & 4 & 17 \end{bmatrix} $$

The inverse of is

Upstudy AI Solution

Answer

Solution

Step 1: Calculate the Determinant of

Step 2: Calculate the Matrix of Cofactors

Step 3: Transpose the Matrix of Cofactors (Adjugate)

Step 4: Calculate the Inverse of

Conclusion

Beyond the Answer

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