Consider the following.
C = 

2	−4	12	

0	6	−12
1	−2	3
 and 
D = 
1
6

1	2	4	

2	1	−4
1	0	−2
,
Find CD.
							
							
							
Find DC.
							
							
							


We are tasked with finding the matrix products \( CD \) and \( DC \), where the given matrices \( C \) and \( D \) are as follows:

\[
C = \begin{bmatrix}
2 & -4 & 12 \\
0 & 6 & -12 \\
1 & -2 & 3
\end{bmatrix}
\quad \text{and} \quad
D = \begin{bmatrix}
1 & 6 & 1 \\
2 & 4 & 2 \\
1 & 0 & -2
\end{bmatrix}
\]

### Step 1: Find \( CD \) (Matrix multiplication \( C \times D \))

Matrix multiplication is performed by taking the dot product of rows of \( C \) with columns of \( D \).

\[
CD = \begin{bmatrix}
(2)(1) + (-4)(2) + (12)(1) & (2)(6) + (-4)(4) + (12)(0) & (2)(1) + (-4)(2) + (12)(-2) \\
(0)(1) + (6)(2) + (-12)(1) & (0)(6) + (6)(4) + (-12)(0) & (0)(1) + (6)(2) + (-12)(-2) \\
(1)(1) + (-2)(2) + (3)(1) & (1)(6) + (-2)(4) + (3)(0) & (1)(1) + (-2)(2) + (3)(-2)
\end{bmatrix}
\]

Now, calculate each element:

\[
CD = \begin{bmatrix}
2 - 8 + 12 & 12 - 16 + 0 & 2 - 8 - 24 \\
0 + 12 - 12 & 0 + 24 + 0 & 0 + 12 + 24 \\
1 - 4 + 3 & 6 - 8 + 0 & 1 - 4 - 6
\end{bmatrix}
= \begin{bmatrix}
6 & -4 & -30 \\
0 & 24 & 36 \\
0 & -2 & -9
\end{bmatrix}
\]

### Step 2: Find \( DC \) (Matrix multiplication \( D \times C \))

Now, calculate \( D \times C \) by multiplying rows of \( D \) by columns of \( C \).

\[
DC = \begin{bmatrix}
(1)(2) + (6)(0) + (1)(1) & (1)(-4) + (6)(6) + (1)(-2) & (1)(12) + (6)(-12) + (1)(3) \\
(2)(2) + (4)(0) + (2)(1) & (2)(-4) + (4)(6) + (2)(-2) & (2)(12) + (4)(-12) + (2)(3) \\
(1)(2) + (0)(0) + (-2)(1) & (1)(-4) + (0)(6) + (-2)(-2) & (1)(12) + (0)(-12) + (-2)(3)
\end{bmatrix}
\]

Now, calculate each element:

\[
DC = \begin{bmatrix}
2 + 0 + 1 & -4 + 36 - 2 & 12 - 72 + 3 \\
4 + 0 + 2 & -8 + 24 - 4 & 24 - 48 + 6 \\
2 + 0 - 2 & -4 + 0 + 4 & 12 + 0 - 6
\end{bmatrix}
= \begin{bmatrix}
3 & 30 & -57 \\
6 & 12 & -18 \\
0 & 0 & 6
\end{bmatrix}
\]

### Final Answers:
\[
CD = \begin{bmatrix}
6 & -4 & -30 \\
0 & 24 & 36 \\
0 & -2 & -9
\end{bmatrix}
\quad \text{and} \quad
DC = \begin{bmatrix}
3 & 30 & -57 \\
6 & 12 & -18 \\
0 & 0 & 6
\end{bmatrix}
\]

Would you like any further clarifications or detailed explanations?

---

Here are five related questions to expand your understanding:

1. What are the conditions for matrix multiplication to be valid?
2. How does the size of the resulting matrix in a multiplication depend on the original matrices?
3. What are the key differences between \( AB \) and \( BA \) for non-square matrices?
4. How does matrix multiplication relate to linear transformations?
5. How can you interpret matrix multiplication in terms of row and column operations?

**Tip:** Matrix multiplication is not commutative, meaning \( AB \neq BA \) in most cases!

Consider the following.
C = 
2	−4	12	
0	6	−12
1	−2	3
and D = 
1
6
1	2	4	
2	1	−4
1	0	−2,
Find CD.
Find DC.

Learn how to find the matrix products CD and DC for given 3x3 matrices C and D. This problem covers key concepts of matrix multiplication in linear algebra, including the dot product of rows and columns. Suitable for students in Grades 11-12.

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