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.