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The image provided contains a problem about matrices, specifically asking to find the inverse of the sum of two matrices, \( P + Q \).

Here is the content of the image:

1. Matrix \( P \) is given as:
   \[
   P = \begin{pmatrix} 1 & 3 \\ -2 & 1 \end{pmatrix}
   \]

2. Matrix \( Q \) is given as:
   \[
   Q = \begin{pmatrix} 2 & 7 \\ 3 & 1 \end{pmatrix}
   \]

3. The problem asks to calculate the inverse of the matrix sum \( P + Q \), i.e., \( \text{Inverse}(P + Q) \).

### Solution Approach:

1. **Calculate \( P + Q \)**: First, add the two matrices element-wise.
   
   \[
   P + Q = \begin{pmatrix} 1+2 & 3+7 \\ -2+3 & 1+1 \end{pmatrix}
   = \begin{pmatrix} 3 & 10 \\ 1 & 2 \end{pmatrix}
   \]

2. **Find the inverse of \( P + Q \)**: To find the inverse of a \( 2 \times 2 \) matrix, we use the formula:
   \[
   A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
   \]
   for a matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), where \( \det(A) = ad - bc \).

   In our case, \( A = P + Q = \begin{pmatrix} 3 & 10 \\ 1 & 2 \end{pmatrix} \).

   So, the determinant \( \det(P + Q) \) is:
   \[
   \det(P + Q) = (3)(2) - (10)(1) = 6 - 10 = -4
   \]

3. **Calculate the inverse**:
   Using the inverse formula, we get:
   \[
   (P + Q)^{-1} = \frac{1}{-4} \begin{pmatrix} 2 & -10 \\ -1 & 3 \end{pmatrix}
   = \begin{pmatrix} -\frac{1}{2} & \frac{5}{2} \\ \frac{1}{4} & -\frac{3}{4} \end{pmatrix}
   \]

Thus, the inverse of \( P + Q \) is:
\[
(P + Q)^{-1} = \begin{pmatrix} -\frac{1}{2} & \frac{5}{2} \\ \frac{1}{4} & -\frac{3}{4} \end{pmatrix}
\]

Would you like further explanation on any step? Here are some related questions:

1. How is the determinant of a matrix calculated?
2. Why does the inverse formula work for \( 2 \times 2 \) matrices?
3. What is the significance of the determinant when finding a matrix's inverse?
4. Can every matrix have an inverse?
5. How does matrix addition affect the overall properties like invertibility?

**Tip**: Always check the determinant before attempting to find the inverse. If the determinant is zero, the matrix has no inverse (it's singular).

Find the inverse of the matrix (P + Q), where P = [[1, 3], [-2, 1]] and Q = [[2, 7], [3, 1]].

Learn how to find the inverse of the matrix sum P + Q, where P and Q are given 2x2 matrices. This problem covers matrix addition and the steps to calculate the inverse using the determinant formula.

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