If 

K

=

1

–

3

–

2

–

5

–

1

1

–

3

–

1

7

 and 

L

=

7

1

–

4

0

2

4

–

2

7

2

,

 what is 

K

L

?

To find the product \( KL \) of the matrices \( K \) and \( L \), we'll multiply them using the matrix multiplication rule, just as before.

Given matrices:
\[
K = \begin{pmatrix}
1 & -3 & -2 \\
-5 & -1 & 1 \\
-3 & -1 & 7
\end{pmatrix}
\]
and
\[
L = \begin{pmatrix}
7 & 1 & -4 \\
0 & 2 & 4 \\
-2 & 7 & 2
\end{pmatrix}
\]

The resulting matrix \( KL \) will be a \( 3 \times 3 \) matrix. Each element \( c_{ij} \) of \( KL \) is calculated as:
\[
c_{ij} = \sum_{k=1}^{3} K_{ik} \times L_{kj}
\]

Let's compute each element step by step.

### First row of KL:
\[
c_{11} = (1)(7) + (-3)(0) + (-2)(-2) = 7 + 0 + 4 = 11
\]
\[
c_{12} = (1)(1) + (-3)(2) + (-2)(7) = 1 - 6 - 14 = -19
\]
\[
c_{13} = (1)(-4) + (-3)(4) + (-2)(2) = -4 - 12 - 4 = -20
\]

### Second row of KL:
\[
c_{21} = (-5)(7) + (-1)(0) + (1)(-2) = -35 + 0 - 2 = -37
\]
\[
c_{22} = (-5)(1) + (-1)(2) + (1)(7) = -5 - 2 + 7 = 0
\]
\[
c_{23} = (-5)(-4) + (-1)(4) + (1)(2) = 20 - 4 + 2 = 18
\]

### Third row of KL:
\[
c_{31} = (-3)(7) + (-1)(0) + (7)(-2) = -21 + 0 - 14 = -35
\]
\[
c_{32} = (-3)(1) + (-1)(2) + (7)(7) = -3 - 2 + 49 = 44
\]
\[
c_{33} = (-3)(-4) + (-1)(4) + (7)(2) = 12 - 4 + 14 = 22
\]

Thus, the product matrix \( KL \) is:
\[
KL = \begin{pmatrix}
11 & -19 & -20 \\
-37 & 0 & 18 \\
-35 & 44 & 22
\end{pmatrix}
\]

Would you like more details, or do you have any questions?

---

1. How can matrix multiplication be applied in physics?
2. What is the inverse of a matrix, and how is it related to matrix multiplication?
3. Can matrix multiplication be used to solve systems of linear equations?
4. How do you determine if two matrices can be multiplied?
5. What are some real-world applications of matrix multiplication?
6. What is a scalar multiplication of a matrix?
7. How does matrix multiplication differ from element-wise multiplication?
8. Can you explain the concept of the transpose of a matrix?

**Tip:** When multiplying matrices, ensure the number of columns in the first matrix equals the number of rows in the second matrix; otherwise, the multiplication is undefined.

Explore how to calculate the product KL of matrices K and L using matrix multiplication. This problem involves detailed steps and calculations for a 3x3 matrix multiplication. Suitable for advanced high school students and beyond.

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