The question is asking about the matrices $I$ and $A$ and whether certain matrix multiplications hold. Here's the breakdown:

You are given two matrices:

• Identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
• Matrix $A = \begin{pmatrix} 8 & 3 \\ 2 & 4 \end{pmatrix}$

Part (a):

You are asked to find:

1. $I \cdot A$
2. $A \cdot I$

Let's compute them one by one:

1. $I \cdot A$:

$I \cdot A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 8 & 3 \\ 2 & 4 \end{pmatrix}$ Multiplying: $I \cdot A = \begin{pmatrix} (1 \cdot 8 + 0 \cdot 2) & (1 \cdot 3 + 0 \cdot 4) \\ (0 \cdot 8 + 1 \cdot 2) & (0 \cdot 3 + 1 \cdot 4) \end{pmatrix} = \begin{pmatrix} 8 & 3 \\ 2 & 4 \end{pmatrix}$ So, $I \cdot A = A$.

2. $A \cdot I$:

$A \cdot I = \begin{pmatrix} 8 & 3 \\ 2 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ Multiplying: $A \cdot I = \begin{pmatrix} (8 \cdot 1 + 3 \cdot 0) & (8 \cdot 0 + 3 \cdot 1) \\ (2 \cdot 1 + 4 \cdot 0) & (2 \cdot 0 + 4 \cdot 1) \end{pmatrix} = \begin{pmatrix} 8 & 3 \\ 2 & 4 \end{pmatrix}$ So, $A \cdot I = A$.

Part (b):

You are asked whether: $A \cdot I = I \cdot A = A$

From the calculations in part (a), we found that both $A \cdot I = A$ and $I \cdot A = A$, so: $A \cdot I = I \cdot A = A$

Thus, the answer to part (b) is yes, the relations $A \cdot I = I \cdot A = A$ hold.

Do you have any further questions or need additional details on any of these steps?

Here are 5 related questions to extend your understanding:

1. What are the properties of the identity matrix in matrix multiplication?
2. How does matrix multiplication change if the matrices are not square?
3. Can the identity matrix be used for non-square matrices? How?
4. What happens when you multiply two identity matrices together?
5. How would matrix multiplication differ if you swapped rows and columns in $A$?

Tip: The identity matrix acts like the number 1 in regular multiplication; multiplying any matrix by it leaves the matrix unchanged.