If `A = [(1,2),(3,-2),(-1,0)]and B = [(1,3,2),(4,-1,3)]` then find the order of AB.

Answer

To find the order of the product of two matrices \( A \) and \( B \), we need to follow these steps:

### Step 1: Identify the order of matrix \( A \)
Matrix \( A \) is given as:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & -2 \\ -1 & 0 \end{pmatrix} \]
Matrix \( A \) has 3 rows and 2 columns. Therefore, the order of matrix \( A \) is \( 3 \times 2 \).

### Step 2: Identify the order of matrix \( B \)
Matrix \( B \) is given as:
\[ B = \begin{pmatrix} 1 & 3 & 2 \\ 4 & -1 & 3 \end{pmatrix} \]
Matrix \( B \) has 2 rows and 3 columns. Therefore, the order of matrix \( B \) is \( 2 \times 3 \).

### Step 3: Check the compatibility for multiplication
For two matrices \( A \) and \( B \) to be multiplied, the number of columns in the first matrix \( A \) must be equal to the number of rows in the second matrix \( B \).

Here, the number of columns in \( A \) is 2 (from the order \( 3 \times 2 \)) and the number of rows in \( B \) is also 2 (from the order \( 2 \times 3 \)). Since these are equal, the multiplication \( AB \) is valid.

### Step 4: Determine the order of the product \( AB \)
The order of the resulting matrix \( AB \) will be determined by the number of rows in \( A \) and the number of columns in \( B \).

- The number of rows in \( A \) is 3.
- The number of columns in \( B \) is 3.

Thus, the order of the product \( AB \) is \( 3 \times 3 \).

### Final Answer
The order of \( AB \) is \( 3 \times 3 \).

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