If `A=[(2,1),(1,3)],B=[(3,2,0),(1,0,4)]`, then AB=…………..

Answer

To find the product of matrices \( A \) and \( B \), we will follow the matrix multiplication rules step by step.

Given:
\[ A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \]
\[ B = \begin{pmatrix} 3 & 2 & 0 \\ 1 & 0 & 4 \end{pmatrix} \]

### Step 1: Identify the dimensions of the matrices
Matrix \( A \) is a \( 2 \times 2 \) matrix and matrix \( B \) is a \( 2 \times 3 \) matrix. The product \( AB \) will be a \( 2 \times 3 \) matrix.

**Hint:** The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be valid.

### Step 2: Set up the multiplication
We will multiply each row of matrix \( A \) by each column of matrix \( B \).

### Step 3: Calculate the elements of the resulting matrix \( AB \)

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

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

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

4. **Second Row, First Column:**
\[ (1 \times 3) + (3 \times 1) = 3 + 3 = 6 \]

5. **Second Row, Second Column:**
\[ (1 \times 2) + (3 \times 0) = 2 + 0 = 2 \]

6. **Second Row, Third Column:**
\[ (1 \times 0) + (3 \times 4) = 0 + 12 = 12 \]

### Step 4: Combine the results into the final matrix
Putting all the calculated elements together, we have:
\[ AB = \begin{pmatrix} 7 & 4 & 4 \\ 6 & 2 & 12 \end{pmatrix} \]

### Final Answer
\[ AB = \begin{pmatrix} 7 & 4 & 4 \\ 6 & 2 & 12 \end{pmatrix} \]

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