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We are given the equation \( PQ = 3I \), where \( Q \) is a \( 3 \times 3 \) non-singular matrix, \( P \) is a known \( 3 \times 3 \) matrix, and \( I \) is the identity matrix. We are asked to find \( Q^{-1} \).

### Step-by-step approach:

1. **Given Information**:
   \[
   P = \begin{pmatrix} 
   3 & -6 & 9 \\
   12 & 2 & 5 \\
   10 & -18 & 15 
   \end{pmatrix}
   \]
   and \( PQ = 3I \), where \( I \) is the identity matrix:
   \[
   I = \begin{pmatrix} 
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1 
   \end{pmatrix}.
   \]

2. **Find \( Q^{-1} \)**:
   Since \( PQ = 3I \), we can isolate \( Q \) by multiplying both sides by \( \frac{1}{3}P^{-1} \) from the left:
   \[
   Q = \frac{1}{3} P^{-1}.
   \]
   Therefore, to find \( Q^{-1} \), we need to calculate \( P^{-1} \) and then scale it by \( \frac{1}{3} \).

### Now let's compute the inverse of matrix \( P \).The inverse of matrix \( Q \), denoted as \( Q^{-1} \), is:

\[
Q^{-1} = \frac{1}{3} P^{-1} = 
\begin{pmatrix} 
-0.0407 & 0.0244 & 0.0163 \\
0.0440 & 0.0152 & -0.0315 \\
0.0799 & 0.0020 & -0.0264 
\end{pmatrix}
\]

Let me know if you'd like further details on how the inverse was calculated!

Here are 5 questions that expand on this topic:
1. How can you verify if a matrix is non-singular?
2. What is the relationship between a matrix and its inverse?
3. How is the identity matrix used in solving matrix equations?
4. What is the determinant of the matrix \( P \), and how does it affect its invertibility?
5. Can a matrix with zero determinant have an inverse?

**Tip**: When solving matrix equations like \( PQ = 3I \), always check if the matrix is non-singular (i.e., has an inverse) by calculating its determinant!

If PQ = 3I where Q is a 3x3 non-singular matrix and P = [3 -6 9; 12 2 5; 10 -18 15], find Q⁻¹.

Learn how to solve for the inverse of a matrix Q given PQ = 3I, where P is a 3x3 non-singular matrix. This problem covers matrix inversion, identity matrix properties, and non-singular matrix theorems. Suitable for college-level linear algebra.

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