https://file.aiquickdraw.com/mathbot/file/1726168960966ets8ovof.jpg

We are tasked with calculating the element \( C(1,2) \), where \( C = AB \). Here, \( A \) and \( B \) are matrices, and \( C(1,2) \) refers to the element in the first row and second column of matrix \( C \).

### Given:
Matrix \( A \):
\[
A = \begin{pmatrix} 
1 & -2 & 3 \\
4 & 0 & 7 
\end{pmatrix}
\]
Matrix \( B \):
\[
B = \begin{pmatrix} 
2 & 4 \\
-1 & 3 \\
0 & 2 
\end{pmatrix}
\]

We need to compute the matrix product \( C = AB \), and then extract the element at position \( C(1,2) \).

### Step 1: Matrix multiplication
To multiply \( A \) and \( B \), the element at position \( C(i,j) \) is given by the dot product of the \( i \)-th row of matrix \( A \) and the \( j \)-th column of matrix \( B \).

#### Compute \( C(1,2) \):
The first row of matrix \( A \) is:
\[
(1, -2, 3)
\]
The second column of matrix \( B \) is:
\[
(4, 3, 2)
\]

Now, calculate the dot product:
\[
C(1,2) = (1 \times 4) + (-2 \times 3) + (3 \times 2)
\]
\[
C(1,2) = 4 + (-6) + 6 = 4
\]

### Final Answer:
The value of \( C(1,2) \) is **4**.

---

Would you like any further explanation or details? Here are some related questions:

1. How is the entire matrix \( C \) formed after multiplying \( A \) and \( B \)?
2. What are the properties of matrix multiplication (e.g., commutativity)?
3. How do we determine the size of the resulting matrix in matrix multiplication?
4. What are the steps to perform matrix multiplication in larger matrices?
5. Can matrix multiplication be applied to transformations in 3D space?

**Tip**: Always verify that the dimensions of matrices allow multiplication; the number of columns in the first matrix must equal the number of rows in the second.

If C = AB, what is C(1,2)? Given matrices A = [[1, -2, 3], [4, 0, 7]] and B = [[2, 4], [-1, 3], [0, 2]].

Learn how to compute the element C(1,2) in matrix multiplication where C = AB, with A = [[1, -2, 3], [4, 0, 7]] and B = [[2, 4], [-1, 3], [0, 2]]. This step-by-step guide covers core concepts in linear algebra, including matrix multiplication and its properties. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation