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!