https://file.aiquickdraw.com/mathbot/file/17253802364233mqwnykl.jpg

To find the expression \( 3A + 2B \), where the matrices \( A \) and \( B \) are provided, we need to perform scalar multiplication and then add the resulting matrices.

Given:

\[
A = \begin{bmatrix} 3 & 4 \\ 6 & -2 \\ 1 & 0 \end{bmatrix}
\quad \text{and} \quad
B = \begin{bmatrix} -3 & 1 \\ 2 & -4 \\ -1 & 5 \end{bmatrix}
\]

### Step 1: Scalar Multiplication
1. Multiply matrix \( A \) by 3:
   \[
   3A = 3 \times \begin{bmatrix} 3 & 4 \\ 6 & -2 \\ 1 & 0 \end{bmatrix}
   = \begin{bmatrix} 9 & 12 \\ 18 & -6 \\ 3 & 0 \end{bmatrix}
   \]

2. Multiply matrix \( B \) by 2:
   \[
   2B = 2 \times \begin{bmatrix} -3 & 1 \\ 2 & -4 \\ -1 & 5 \end{bmatrix}
   = \begin{bmatrix} -6 & 2 \\ 4 & -8 \\ -2 & 10 \end{bmatrix}
   \]

### Step 2: Matrix Addition
Now, add the matrices \( 3A \) and \( 2B \):
\[
3A + 2B = \begin{bmatrix} 9 & 12 \\ 18 & -6 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} -6 & 2 \\ 4 & -8 \\ -2 & 10 \end{bmatrix}
\]

Perform the element-wise addition:
\[
3A + 2B = \begin{bmatrix} 9 + (-6) & 12 + 2 \\ 18 + 4 & -6 + (-8) \\ 3 + (-2) & 0 + 10 \end{bmatrix}
= \begin{bmatrix} 3 & 14 \\ 22 & -14 \\ 1 & 10 \end{bmatrix}
\]

### Final Answer:
\[
3A + 2B = \begin{bmatrix} 3 & 14 \\ 22 & -14 \\ 1 & 10 \end{bmatrix}
\]

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would you find the determinant of matrix \( A \)?
2. What is the transpose of matrix \( B \)?
3. Can matrix \( A \) and \( B \) be multiplied? If so, what is the product \( AB \)?
4. How would you find the inverse of matrix \( A \) if it exists?
5. What are the eigenvalues and eigenvectors of matrix \( A \)?

**Tip:** When performing matrix operations, ensure that the dimensions of the matrices are compatible for the operation you wish to carry out.

Find 3A + 2B for the given matrices A and B.

Solve the matrix algebra problem of finding 3A + 2B for the given matrices A and B. The solution involves scalar multiplication and matrix addition, key concepts in matrix algebra. 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