D = 

2	8	

3	9
      E = 

1	0	3	

6	1	0
Find DE.

We are asked to multiply two matrices \( D \) and \( E \):

Matrix \( D \):
\[
D = \begin{pmatrix} 
2 & 8 \\ 
3 & 9 
\end{pmatrix}
\]

Matrix \( E \):
\[
E = \begin{pmatrix} 
1 & 0 & 3 \\ 
6 & 1 & 0 
\end{pmatrix}
\]

Matrix multiplication is defined as follows: If matrix \( D \) is of size \( m \times n \), and matrix \( E \) is of size \( n \times p \), their product \( DE \) will be of size \( m \times p \). Here, \( D \) is a \( 2 \times 2 \) matrix, and \( E \) is a \( 2 \times 3 \) matrix, so their product \( DE \) will be a \( 2 \times 3 \) matrix.

To calculate the element at position \( (i, j) \) of the product, we take the dot product of the \( i \)-th row of matrix \( D \) and the \( j \)-th column of matrix \( E \).

Let's compute \( DE \):

### First row of \( DE \):
1. \( (1, 1) \)-th element: \( 2 \times 1 + 8 \times 6 = 2 + 48 = 50 \)
2. \( (1, 2) \)-th element: \( 2 \times 0 + 8 \times 1 = 0 + 8 = 8 \)
3. \( (1, 3) \)-th element: \( 2 \times 3 + 8 \times 0 = 6 + 0 = 6 \)

### Second row of \( DE \):
1. \( (2, 1) \)-th element: \( 3 \times 1 + 9 \times 6 = 3 + 54 = 57 \)
2. \( (2, 2) \)-th element: \( 3 \times 0 + 9 \times 1 = 0 + 9 = 9 \)
3. \( (2, 3) \)-th element: \( 3 \times 3 + 9 \times 0 = 9 + 0 = 9 \)

Thus, the product matrix \( DE \) is:
\[
DE = \begin{pmatrix} 
50 & 8 & 6 \\ 
57 & 9 & 9 
\end{pmatrix}
\]

Would you like a detailed explanation of matrix multiplication or have any questions?

Here are 5 follow-up questions you might find interesting:
1. How do you calculate the dot product in a matrix multiplication?
2. What conditions must hold for two matrices to be multiplied?
3. Can you find the inverse of a matrix, and when does it exist?
4. How is matrix multiplication used in solving systems of equations?
5. How does matrix multiplication relate to transformations in geometry?

**Tip**: When multiplying matrices, ensure the number of columns in the first matrix equals the number of rows in the second matrix!

D = 

2 8

3 9

E = 

1 0 3

6 1 0

Find DE.

Learn how to multiply a 2x2 matrix by a 2x3 matrix with a detailed step-by-step solution. This problem demonstrates key concepts in linear algebra, such as matrix multiplication, and is suitable for students in Grades 9-12.

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