https://file.aiquickdraw.com/mathbot/file/1723678944156tu3hhyvq.jpg

Given the following matrices.
[
G
]
=

9	-1	0	6
-3	5	3	1
2	-8	10	-2	 	and
[
C
]
=

10	9	-9	0
-2	1	-7	2
-5	6	8	-1
-4	-6	5	-10

Find
[
G
]
⋅
[
C
]
T
=

To solve the problem of finding the product of matrix \( G \) and the transpose of matrix \( C \), denoted as \( G \cdot C^T \), follow these steps:

### 1. Write Down the Matrices

Given matrices \( G \) and \( C \):

\[
G =
\begin{bmatrix}
9 & -1 & 0 & 6 \\
-3 & 5 & 3 & 1 \\
2 & -8 & 10 & -2
\end{bmatrix}
\]

\[
C =
\begin{bmatrix}
10 & 9 & -9 & 0 \\
-2 & 1 & -7 & 2 \\
-5 & 6 & 8 & -1 \\
-4 & -6 & 5 & -10
\end{bmatrix}
\]

### 2. Find the Transpose of Matrix \( C \)

The transpose of matrix \( C \), denoted \( C^T \), is obtained by swapping the rows and columns of \( C \):

\[
C^T =
\begin{bmatrix}
10 & -2 & -5 & -4 \\
9 & 1 & 6 & -6 \\
-9 & -7 & 8 & 5 \\
0 & 2 & -1 & -10
\end{bmatrix}
\]

### 3. Multiply Matrix \( G \) by Matrix \( C^T \)

To find \( G \cdot C^T \), multiply each row of \( G \) by each column of \( C^T \).

- **First Row of \( G \) with all Columns of \( C^T \):**

\[
\text{First element} = (9)(10) + (-1)(9) + (0)(-9) + (6)(0) = 90 - 9 + 0 + 0 = 81
\]

\[
\text{Second element} = (9)(-2) + (-1)(1) + (0)(-7) + (6)(2) = -18 - 1 + 0 + 12 = -7
\]

\[
\text{Third element} = (9)(-5) + (-1)(6) + (0)(8) + (6)(-1) = -45 - 6 + 0 - 6 = -57
\]

\[
\text{Fourth element} = (9)(-4) + (-1)(-6) + (0)(5) + (6)(-10) = -36 + 6 + 0 - 60 = -90
\]

- **Second Row of \( G \) with all Columns of \( C^T \):**

\[
\text{First element} = (-3)(10) + (5)(9) + (3)(-9) + (1)(0) = -30 + 45 - 27 + 0 = -12
\]

\[
\text{Second element} = (-3)(-2) + (5)(1) + (3)(-7) + (1)(2) = 6 + 5 - 21 + 2 = -8
\]

\[
\text{Third element} = (-3)(-5) + (5)(6) + (3)(8) + (1)(-1) = 15 + 30 + 24 - 1 = 68
\]

\[
\text{Fourth element} = (-3)(-4) + (5)(-6) + (3)(5) + (1)(-10) = 12 - 30 + 15 - 10 = -13
\]

- **Third Row of \( G \) with all Columns of \( C^T \):**

\[
\text{First element} = (2)(10) + (-8)(9) + (10)(-9) + (-2)(0) = 20 - 72 - 90 + 0 = -142
\]

\[
\text{Second element} = (2)(-2) + (-8)(1) + (10)(-7) + (-2)(2) = -4 - 8 - 70 - 4 = -86
\]

\[
\text{Third element} = (2)(-5) + (-8)(6) + (10)(8) + (-2)(-1) = -10 - 48 + 80 + 2 = 24
\]

\[
\text{Fourth element} = (2)(-4) + (-8)(-6) + (10)(5) + (-2)(-10) = -8 + 48 + 50 + 20 = 110
\]

### 4. Write Down the Resulting Matrix

The resulting matrix \( G \cdot C^T \) is:

\[
G \cdot C^T =
\begin{bmatrix}
81 & -7 & -57 & -90 \\
-12 & -8 & 68 & -13 \\
-142 & -86 & 24 & 110
\end{bmatrix}
\]

### Conclusion
So, the product of matrix \( G \) and the transpose of matrix \( C \) is:

\[
\begin{bmatrix}
81 & -7 & -57 & -90 \\
-12 & -8 & 68 & -13 \\
-142 & -86 & 24 & 110
\end{bmatrix}
\]

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

---

Here are some related questions to consider:
1. What is matrix multiplication, and how does it work?
2. How do you find the transpose of a matrix?
3. What are the properties of matrix multiplication?
4. Can matrix multiplication be commutative?
5. How do you verify the correctness of a matrix multiplication?
6. What are the applications of matrix multiplication in real-world problems?
7. How does the dimension of matrices affect the multiplication process?
8. How can matrix operations be implemented in programming languages like Python?

**Tip:** Always check the dimensions of the matrices before multiplying them to ensure that the operation is valid.

Learn how to multiply matrix G by the transpose of matrix C step-by-step. This problem involves matrix operations and the transpose concept. Suitable for advanced high school or college-level students.

Math Problem Statement

Given the following matrices. [ G ]

9 -1 0 6 -3 5 3 1 2 -8 10 -2 and [ C ]

Find [ G ] ⋅ [ C ] T

Solution

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