The following represent matrices and the dimension of each is also stated below.
A is a matrix of size 4X2
B is a matrix of size 3X5
C is a matrix of size 7X2
D is a matrix of size 5X4
E is a matrix of size 5X3
1. The product AD is defined.
a. True
b. False
2. The product DB is defined.
a. True
b. False
3. The product AE is defined.
a. True
b. False
4. The product BE is defined. The dimension of the product is
a. 1X1
b. 2X3
c. 3X5
d. 3X3
e. 5X5 5. The product BD is defined. The dimension of the product is
a. 4X4
b. 4X3
c. 5X5
d. 3X3
e. 3X4  

Let's analyze the matrix multiplication problems based on the given dimensions:

- **A**: 4 × 2
- **B**: 3 × 5
- **C**: 7 × 2
- **D**: 5 × 4
- **E**: 5 × 3

### Key Rule for Matrix Multiplication:
To multiply two matrices \(X\) (of dimension \(m \times n\)) and \(Y\) (of dimension \(p \times q\)), the number of columns of \(X\) (i.e., \(n\)) must equal the number of rows of \(Y\) (i.e., \(p\)). The resulting product matrix will have dimensions \(m \times q\).

Now let's solve each question:

1. **The product AD is defined.**
   - **A**: 4 × 2
   - **D**: 5 × 4
   - For the product to be defined, the number of columns of \(A\) (2) must match the number of rows of \(D\) (5). This is not true.
   - **Answer**: **b. False**

2. **The product DB is defined.**
   - **D**: 5 × 4
   - **B**: 3 × 5
   - For the product to be defined, the number of columns of \(D\) (4) must match the number of rows of \(B\) (3). This is not true.
   - **Answer**: **b. False**

3. **The product AE is defined.**
   - **A**: 4 × 2
   - **E**: 5 × 3
   - For the product to be defined, the number of columns of \(A\) (2) must match the number of rows of \(E\) (5). This is not true.
   - **Answer**: **b. False**

4. **The product BE is defined. The dimension of the product is**
   - **B**: 3 × 5
   - **E**: 5 × 3
   - For the product to be defined, the number of columns of \(B\) (5) must match the number of rows of \(E\) (5). This is true.
   - The resulting matrix will have dimensions \(3 \times 3\).
   - **Answer**: **d. 3 × 3**

5. **The product BD is defined. The dimension of the product is**
   - **B**: 3 × 5
   - **D**: 5 × 4
   - For the product to be defined, the number of columns of \(B\) (5) must match the number of rows of \(D\) (5). This is true.
   - The resulting matrix will have dimensions \(3 \times 4\).
   - **Answer**: **e. 3 × 4**

### To summarize:
1. **b. False**
2. **b. False**
3. **b. False**
4. **d. 3 × 3**
5. **e. 3 × 4**

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Would you like a detailed explanation of any of the answers or concepts? Here are some additional practice questions for you:

1. Is the product \(AC\) defined? Why or why not?
2. What is the dimension of the product \(CD\) if it is defined?
3. If matrix \(F\) is of size 4 × 5, is the product \(EF\) defined?
4. If matrix \(G\) is of size 2 × 3, is the product \(CG\) defined?
5. What happens if you try to multiply two matrices with incompatible dimensions?

**Tip**: Remember, for matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix!

The following represent matrices and the dimension of each is also stated below.
A is a matrix of size 4X2
B is a matrix of size 3X5
C is a matrix of size 7X2
D is a matrix of size 5X4
E is a matrix of size 5X3
1. The product AD is defined.
a. True
b. False
2. The product DB is defined.
a. True
b. False
3. The product AE is defined.
a. True
b. False
4. The product BE is defined. The dimension of the product is
a. 1X1
b. 2X3
c. 3X5
d. 3X3
e. 5X5
5. The product BD is defined. The dimension of the product is
a. 4X4
b. 4X3
c. 5X5
d. 3X3
e. 3X4

Understand matrix multiplication through the example of matrices A, B, C, D, and E with different dimensions. Learn how to verify whether matrix products like AD, DB, AE, and BE are defined, and calculate their resulting dimensions. This problem covers advanced linear algebra concepts suitable for undergraduate and advanced high school students.

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