Use the following matrices to perform the indicated operation, when possible. (If not possible, enter IMPOSSIBLE into any cell of the matrix.)
A = 

0	1	2	

2	1	6
3	2	0
      B = 

1	1	4	0	

5	2	1	1
2	0	3	1
Find AB.





We are asked to find the product of matrices \(A\) and \(B\), denoted by \(AB\). Matrix multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second matrix. 

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

- Matrix \(A\) is \(3 \times 3\) (3 rows and 3 columns).
- Matrix \(B\) is \(3 \times 4\) (3 rows and 4 columns).

Since matrix \(A\) has 3 columns and matrix \(B\) has 3 rows, matrix multiplication is possible. The resulting matrix \(AB\) will have dimensions \(3 \times 4\) (3 rows from \(A\) and 4 columns from \(B\)).

The formula for matrix multiplication is:
\[
(AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj}
\]
where \(i\) is the row of matrix \(A\), \(j\) is the column of matrix \(B\), and \(k\) sums over the shared dimension.

Let's compute the product \(AB\).

### Step-by-step multiplication:

1. **First row of \(A\) with all columns of \(B\):**
   \[
   \text{Row 1 of } AB = \left( 0 \times 1 + 1 \times 5 + 2 \times 2, \, 0 \times 1 + 1 \times 2 + 2 \times 0, \, 0 \times 4 + 1 \times 1 + 2 \times 3, \, 0 \times 0 + 1 \times 1 + 2 \times 1 \right)
   \]
   \[
   = (9, 2, 7, 3)
   \]

2. **Second row of \(A\) with all columns of \(B\):**
   \[
   \text{Row 2 of } AB = \left( 2 \times 1 + 1 \times 5 + 6 \times 2, \, 2 \times 1 + 1 \times 2 + 6 \times 0, \, 2 \times 4 + 1 \times 1 + 6 \times 3, \, 2 \times 0 + 1 \times 1 + 6 \times 1 \right)
   \]
   \[
   = (24, 4, 29, 7)
   \]

3. **Third row of \(A\) with all columns of \(B\):**
   \[
   \text{Row 3 of } AB = \left( 3 \times 1 + 2 \times 5 + 0 \times 2, \, 3 \times 1 + 2 \times 2 + 0 \times 0, \, 3 \times 4 + 2 \times 1 + 0 \times 3, \, 3 \times 0 + 2 \times 1 + 0 \times 1 \right)
   \]
   \[
   = (13, 7, 14, 2)
   \]

### Final Result:
\[
AB = \begin{pmatrix}
9 & 2 & 7 & 3 \\
24 & 4 & 29 & 7 \\
13 & 7 & 14 & 2
\end{pmatrix}
\]

Would you like any additional details on how these calculations were performed?

### 5 Related Questions:
1. How do you verify if two matrices can be multiplied?
2. What is the significance of matrix multiplication in real-world applications?
3. How can you find the determinant of a matrix?
4. What are the properties of matrix transposition?
5. How do you find the inverse of a matrix?

### Tip:
Always check the dimensions of matrices before attempting to multiply them to ensure compatibility for matrix multiplication.

Use the following matrices to perform the indicated operation, when possible. (If not possible, enter IMPOSSIBLE into any cell of the matrix.) A = 0 1 2 2 1 6 3 2 0 B = 1 1 4 0 5 2 1 1 2 0 3 1 Find AB.

Learn how to perform matrix multiplication between a 3x3 matrix A and a 3x4 matrix B. This problem includes a step-by-step guide, covering core concepts in matrix operations and linear algebra. Suitable for high school students in Grades 10-12.

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