https://file.aiquickdraw.com/mathbot/file/1730130807279e9fvygub.jpg

To determine if the matrix products \( AB \) and \( BA \) are defined, we need to check the dimensions of matrices \( A \) and \( B \) and see if they align properly for multiplication.

### Given Matrices
Matrix \( A \) is:
\[
A = \begin{bmatrix} -3 & 1 \\ 4 & 0 \end{bmatrix}
\]
Matrix \( B \) is:
\[
B = \begin{bmatrix} 3 & -5 \\ -3 & 9 \\ 8 & -2 \end{bmatrix}
\]

### Step 1: Determine Dimensions
- Matrix \( A \) has dimensions \( 2 \times 2 \) (2 rows and 2 columns).
- Matrix \( B \) has dimensions \( 3 \times 2 \) (3 rows and 2 columns).

### Step 2: Check for Compatibility of \( AB \) and \( BA \)
1. **For \( AB \) to be defined**: The number of columns in \( A \) (2) must match the number of rows in \( B \) (3). Since this is not the case, the product \( AB \) is **not defined**.

2. **For \( BA \) to be defined**: The number of columns in \( B \) (2) must match the number of rows in \( A \) (2). This condition is satisfied, so the product \( BA \) **is defined** and will result in a \( 3 \times 2 \) matrix.

Thus, the correct answer is:
- **B**: The product \( AB \) is not defined.

Would you like further details or an explanation of how to calculate \( BA \)?

---

### Related Questions
1. What are the general conditions for matrix multiplication to be defined?
2. How do you calculate the product of two matrices?
3. What properties does matrix multiplication have regarding commutativity?
4. How can we verify the dimensions of the resulting matrix after multiplication?
5. In what scenarios would both \( AB \) and \( BA \) be defined?

### Tip
Remember, matrix multiplication is not commutative: \( AB \neq BA \) in general.

Two matrices A and B are given. Calculate whichever of the matrices AB and BA is defined. Given matrices: A = [[-3, 4]], B = [[3, -5], [-3, 9], [8, -2]]. Find AB and determine if the product is defined or not.

Learn how to determine if matrix products AB or BA are defined for matrices A = [[-3, 4]] and B = [[3, -5], [-3, 9], [8, -2]]. This problem covers key linear algebra concepts and matrix multiplication rules. Suitable for high school 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