https://file.aiquickdraw.com/mathbot/file/1727233241111cc39i42e.jpg

To solve the matrix expression in the image, we need to multiply the two matrices on the left and then add the result to the matrix on the right. Here's the step-by-step solution:

### Step 1: Matrix multiplication
We are asked to multiply the two matrices:

\[
\begin{pmatrix}
3 & 3 \\
-4 & -3 \\
1 & -2
\end{pmatrix}
\times
\begin{pmatrix}
-1 & -3 \\
-3 & -1
\end{pmatrix}
\]

This is a 3x2 matrix multiplied by a 2x2 matrix, resulting in a 3x2 matrix.

Let's compute it:

\[
C_{ij} = \sum_{k} A_{ik} B_{kj}
\]

\[
C_{11} = (3 \times -1) + (3 \times -3) = -3 + (-9) = -12
\]
\[
C_{12} = (3 \times -3) + (3 \times -1) = -9 + (-3) = -12
\]
\[
C_{21} = (-4 \times -1) + (-3 \times -3) = 4 + (-9) = -5
\]
\[
C_{22} = (-4 \times -3) + (-3 \times -1) = 12 + 3 = 15
\]
\[
C_{31} = (1 \times -1) + (-2 \times -3) = -1 + 6 = 5
\]
\[
C_{32} = (1 \times -3) + (-2 \times -1) = -3 + 2 = -1
\]

So the result of the matrix multiplication is:

\[
\begin{pmatrix}
-12 & -12 \\
-5 & 15 \\
5 & -1
\end{pmatrix}
\]

### Step 2: Matrix addition
Now, add this result to the given matrix on the right:

\[
\begin{pmatrix}
-12 & -12 \\
-5 & 15 \\
5 & -1
\end{pmatrix}
+
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
5 & 6
\end{pmatrix}
\]

The matrix addition is done element-wise:

\[
\begin{pmatrix}
-12 + 1 & -12 + 2 \\
-5 + 3 & 15 + 4 \\
5 + 5 & -1 + 6
\end{pmatrix}
=
\begin{pmatrix}
-11 & -10 \\
-2 & 19 \\
10 & 5
\end{pmatrix}
\]

### Final answer:
The final result of the expression is:

\[
\begin{pmatrix}
-11 & -10 \\
-2 & 19 \\
10 & 5
\end{pmatrix}
\]

Would you like further clarification or additional details on any of these steps?

### 5 Related Questions:
1. How do you determine if two matrices can be multiplied?
2. What are the properties of matrix multiplication?
3. How do you calculate the determinant of a matrix?
4. Can matrix multiplication be commutative?
5. What are the applications of matrix multiplication in real-life scenarios?

### Tip:
When performing matrix multiplication, always check the dimensions to ensure the operation is valid (the number of columns in the first matrix must equal the number of rows in the second matrix).

Multiply the two matrices [[3, 3], [-4, -3], [1, -2]] and [[-1, -3], [-3, -1]], then add the result to the matrix [[1, 2], [3, 4], [5, 6]].

Solve a matrix multiplication and addition problem involving two 3x2 and 2x2 matrices. This solution explains step-by-step how to perform matrix multiplication and addition, suitable for high school students in Grades 10-12.

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