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Let's focus on solving question 22 from the image.

The question reads:
**Given matrices:**
\[
M = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 1 \\ 0 & 1 & 1 \end{pmatrix}, \quad N = \begin{pmatrix} 5 & 0 \\ 1 & 4 \\ 1 & 4 \end{pmatrix}
\]

**Find matrices M × N and N × M.**

### Solution:

#### Step 1: Multiplication of \( M \times N \)

Matrix \( M \) is of size \( 3 \times 3 \), and matrix \( N \) is of size \( 3 \times 2 \). For the matrix product \( M \times N \), the resulting matrix will be of size \( 3 \times 2 \) (the number of rows in \( M \) by the number of columns in \( N \)).

\[
M = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 1 \\ 0 & 1 & 1 \end{pmatrix}, \quad N = \begin{pmatrix} 5 & 0 \\ 1 & 4 \\ 1 & 4 \end{pmatrix}
\]

We compute the product \( M \times N \):

\[
M \times N = \begin{pmatrix} (1 \cdot 5 + 2 \cdot 1 + 3 \cdot 1) & (1 \cdot 0 + 2 \cdot 4 + 3 \cdot 4) \\ 
(2 \cdot 5 + 4 \cdot 1 + 1 \cdot 1) & (2 \cdot 0 + 4 \cdot 4 + 1 \cdot 4) \\ 
(0 \cdot 5 + 1 \cdot 1 + 1 \cdot 1) & (0 \cdot 0 + 1 \cdot 4 + 1 \cdot 4) \end{pmatrix}
\]

Calculating element-wise:

\[
M \times N = \begin{pmatrix} (5 + 2 + 3) & (0 + 8 + 12) \\ 
(10 + 4 + 1) & (0 + 16 + 4) \\ 
(0 + 1 + 1) & (0 + 4 + 4) \end{pmatrix}
\]

\[
M \times N = \begin{pmatrix} 10 & 20 \\ 15 & 20 \\ 2 & 8 \end{pmatrix}
\]

Thus, \( M \times N = \begin{pmatrix} 10 & 20 \\ 15 & 20 \\ 2 & 8 \end{pmatrix} \).

#### Step 2: Multiplication of \( N \times M \)

Matrix \( N \) is of size \( 3 \times 2 \), and matrix \( M \) is of size \( 3 \times 3 \). The multiplication \( N \times M \) is not defined, because the number of columns in \( N \) does not match the number of rows in \( M \) (they are not conformable for multiplication).

Thus, \( N \times M \) is **not possible**.

### Final Answer:

- \( M \times N = \begin{pmatrix} 10 & 20 \\ 15 & 20 \\ 2 & 8 \end{pmatrix} \)
- \( N \times M \) is not possible.

Would you like more details or have any questions? Here are five related questions to explore further:

1. How do you multiply two matrices with different dimensions?
2. What are the necessary conditions for matrix multiplication to be valid?
3. Can matrix multiplication be commutative? Why or why not?
4. How would the result differ if \( M \) and \( N \) had different elements?
5. What is the role of the identity matrix in matrix multiplication?

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

Find matrices M × N and N × M, where M = [[1, 2, 3], [2, 4, 1], [0, 1, 1]] and N = [[5, 0], [1, 4], [1, 4]].

Learn how to find the product of two matrices M and N with detailed steps. Explore the properties of matrix multiplication and understand why N × M is not possible. This problem is suitable for students in Grades 11-12 and covers key concepts in matrix algebra.

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