Find the inverse of the following Matrix:


1 2 3
0 1 4
5 6 0

To find the inverse of a 3x3 matrix, we follow these steps:

Given matrix \( A \):
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\]

### Steps:

1. **Find the determinant** of the matrix \( A \).
   
2. **Find the matrix of minors**.

3. **Find the matrix of cofactors**.

4. **Find the adjugate (transpose of the cofactor matrix)**.

5. **Divide the adjugate matrix by the determinant** to get the inverse.

Let's start with step 1 and calculate the determinant \( \text{det}(A) \).

### Determinant calculation:
\[
\text{det}(A) = 1 \cdot \left(1 \cdot 0 - 4 \cdot 6 \right) - 2 \cdot \left(0 \cdot 0 - 4 \cdot 5 \right) + 3 \cdot \left(0 \cdot 6 - 1 \cdot 5 \right)
\]

Let me calculate this.The determinant of the matrix \( A \) is approximately \( 1 \), which means the matrix is invertible.

Now that we have the determinant, we can proceed to the next steps: finding the matrix of minors, cofactors, adjugate, and the inverse.

Since the determinant is \( 1 \), the inverse will simply be the adjugate matrix.

Let me compute the inverse now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can compute the inverse of the matrix manually by following these steps:

1. **Find the matrix of minors** for the given matrix.
2. **Find the cofactors**.
3. **Transpose the cofactor matrix** (to get the adjugate).
4. **Divide the adjugate matrix by the determinant**, which is 1 in this case.

Would you like a more detailed walkthrough of these steps? 

### Related Questions:
1. What is the matrix of minors for this matrix?
2. How do you compute the cofactor of a matrix?
3. What role does the determinant play in finding an inverse?
4. How would this process change for a 2x2 matrix?
5. What does it mean if the determinant is 0?

**Tip:** Always verify the determinant before attempting to find the inverse of a matrix; if it’s zero, the matrix is non-invertible.

Find the inverse of the following Matrix:
1 2 3
0 1 4
5 6 0

Discover how to find the inverse of a 3x3 matrix with a detailed walkthrough. This problem involves calculating the determinant, minors, cofactors, and adjugate of the matrix. Ideal for undergraduate students studying linear algebra.

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