Let `A=[(2,4),(1,-3)]` and `B=[(1,-1,5),(0,2,6)]`
(a) Find AB. (b) Is BA defined ? Justify your answer.

Answer

To solve the given problem, we will perform the following steps:

### Step 1: Find the product \( AB \)

Given:
- Matrix \( A = \begin{pmatrix} 2 & 4 \\ 1 & -3 \end{pmatrix} \)
- Matrix \( B = \begin{pmatrix} 1 & -1 & 5 \\ 0 & 2 & 6 \end{pmatrix} \)

To find the product \( AB \), we will use the formula for matrix multiplication, which involves taking the dot product of the rows of the first matrix with the columns of the second matrix.

#### Calculation:
1. **First Row, First Column:**
\[ (2 \times 1) + (4 \times 0) = 2 + 0 = 2 \]

2. **First Row, Second Column:**
\[ (2 \times -1) + (4 \times 2) = -2 + 8 = 6 \]

3. **First Row, Third Column:**
\[ (2 \times 5) + (4 \times 6) = 10 + 24 = 34 \]

4. **Second Row, First Column:**
\[ (1 \times 1) + (-3 \times 0) = 1 + 0 = 1 \]

5. **Second Row, Second Column:**
\[ (1 \times -1) + (-3 \times 2) = -1 - 6 = -7 \]

6. **Second Row, Third Column:**
\[ (1 \times 5) + (-3 \times 6) = 5 - 18 = -13 \]

Putting it all together, we have:
\[ AB = \begin{pmatrix} 2 & 6 & 34 \\ 1 & -7 & -13 \end{pmatrix} \]

### Step 2: Check if \( BA \) is defined

To check if \( BA \) is defined, we need to look at the dimensions of the matrices:
- Matrix \( B \) is \( 2 \times 3 \)
- Matrix \( A \) is \( 2 \times 2 \)

For matrix multiplication \( BA \) to be defined, the number of columns in \( B \) must equal the number of rows in \( A \).

#### Dimensions:
- \( B \) has 3 columns.
- \( A \) has 2 rows.

Since the number of columns in \( B \) (3) does not equal the number of rows in \( A \) (2), the product \( BA \) is **not defined**.

### Final Answer:
(a) \( AB = \begin{pmatrix} 2 & 6 & 34 \\ 1 & -7 & -13 \end{pmatrix} \)

(b) \( BA \) is **not defined** because the number of columns in \( B \) (3) does not match the number of rows in \( A \) (2).

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