If G=
–
8	0
–
8
–
3	9	7
7
–
3
–
8
and H=
–
5	6
–
8
3	0	3
1
–
6	1
, what is GH?

To find the product \( GH \) of two matrices \( G \) and \( H \), we will multiply them using the matrix multiplication rule. The product of two matrices is calculated by taking the dot product of rows from the first matrix with columns of the second matrix.

Given:
\[ 
G = \begin{pmatrix}
-8 & 0 & -8 \\
-3 & 9 & 7 \\
7 & -3 & -8
\end{pmatrix}
\]
and
\[ 
H = \begin{pmatrix}
-5 & 6 & -8 \\
3 & 0 & 3 \\
1 & -6 & 1
\end{pmatrix}
\]

The resulting matrix \( GH \) will be a \( 3 \times 3 \) matrix. Each element \( c_{ij} \) of \( GH \) is calculated as:

\[
c_{ij} = \sum_{k=1}^{3} G_{ik} \times H_{kj}
\]

Let's compute each element step by step.

### First row of GH:
\[
c_{11} = (-8)(-5) + (0)(3) + (-8)(1) = 40 + 0 - 8 = 32
\]
\[
c_{12} = (-8)(6) + (0)(0) + (-8)(-6) = -48 + 0 + 48 = 0
\]
\[
c_{13} = (-8)(-8) + (0)(3) + (-8)(1) = 64 + 0 - 8 = 56
\]

### Second row of GH:
\[
c_{21} = (-3)(-5) + (9)(3) + (7)(1) = 15 + 27 + 7 = 49
\]
\[
c_{22} = (-3)(6) + (9)(0) + (7)(-6) = -18 + 0 - 42 = -60
\]
\[
c_{23} = (-3)(-8) + (9)(3) + (7)(1) = 24 + 27 + 7 = 58
\]

### Third row of GH:
\[
c_{31} = (7)(-5) + (-3)(3) + (-8)(1) = -35 - 9 - 8 = -52
\]
\[
c_{32} = (7)(6) + (-3)(0) + (-8)(-6) = 42 + 0 + 48 = 90
\]
\[
c_{33} = (7)(-8) + (-3)(3) + (-8)(1) = -56 - 9 - 8 = -73
\]

Thus, the product matrix \( GH \) is:
\[
GH = \begin{pmatrix}
32 & 0 & 56 \\
49 & -60 & 58 \\
-52 & 90 & -73
\end{pmatrix}
\]

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

---

1. How do you multiply matrices of different dimensions?
2. What are the properties of matrix multiplication?
3. How can matrix multiplication be used in computer graphics?
4. Can you explain the identity matrix and its role in multiplication?
5. How is matrix multiplication used in solving linear equations?
6. What is the significance of the determinant in matrix operations?
7. How does matrix multiplication relate to linear transformations?
8. Can you show an example of matrix multiplication with larger matrices?

**Tip:** Remember, matrix multiplication is not commutative, meaning \( AB \neq BA \) in general!

Explore how to compute the product of matrices G and H, step-by-step. Matrix multiplication involves calculating dot products of rows and columns, illustrated with examples from matrices G and H. Suitable for high school and above.

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