Given two matrices A and B below, calculate the matrix products, AB and BA. Check if determinants of AB and BA are same.
A=(█(3@4)) , and B=■((2&5))


We are given two matrices:

\[
A = \begin{pmatrix} 3 \end{pmatrix}
\quad \text{and} \quad 
B = \begin{pmatrix} 2 & 5 \end{pmatrix}
\]

These matrices are of different sizes:

- \( A \) is a \( 1 \times 1 \) matrix.
- \( B \) is a \( 1 \times 2 \) matrix.

### Step 1: Matrix Product \( AB \)

First, we'll compute the product \( AB \), where matrix \( A \) is multiplied by matrix \( B \).

\[
AB = \begin{pmatrix} 3 \end{pmatrix} \times \begin{pmatrix} 2 & 5 \end{pmatrix}
\]

This is a scalar multiplication because \( A \) is a \( 1 \times 1 \) matrix. The result is:

\[
AB = \begin{pmatrix} 3 \times 2 & 3 \times 5 \end{pmatrix} = \begin{pmatrix} 6 & 15 \end{pmatrix}
\]

Thus, \( AB = \begin{pmatrix} 6 & 15 \end{pmatrix} \).

### Step 2: Matrix Product \( BA \)

Next, we'll compute the product \( BA \), where matrix \( B \) is multiplied by matrix \( A \).

\[
BA = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \times \begin{pmatrix} 3 \end{pmatrix}
\]

Again, this is a scalar multiplication. The result is:

\[
BA = \begin{pmatrix} 2 \times 3 \\ 5 \times 3 \end{pmatrix} = \begin{pmatrix} 6 \\ 15 \end{pmatrix}
\]

Thus, \( BA = \begin{pmatrix} 6 \\ 15 \end{pmatrix} \).

### Step 3: Determinants of \( AB \) and \( BA \)

- \( AB \) is a \( 1 \times 2 \) matrix, and it doesn't have a determinant because determinants are only defined for square matrices.
- \( BA \) is a \( 2 \times 1 \) matrix, and it also doesn't have a determinant for the same reason.

### Conclusion:
Neither \( AB \) nor \( BA \) are square matrices, so their determinants cannot be calculated, and hence, they cannot be compared.

Do you have any questions or want further details?

---

Here are some related questions to explore further:

1. What is the condition for two matrices to be multiplied?
2. Can you explain how to compute the determinant of a \( 2 \times 2 \) matrix?
3. What are the properties of matrix multiplication?
4. How do scalar multiplications work in matrices?
5. When does the product of two matrices give the identity matrix?

**Tip:** A matrix must be square to compute its determinant.

Given two matrices A and B below, calculate the matrix products, AB and BA. Check if determinants of AB and BA are same. A=(█(3@4)) , and B=■((2&5))

Calculate the matrix products AB and BA for given matrices A and B. This problem focuses on matrix multiplication and determinants, explaining why AB and BA do not have determinants since they are non-square matrices. Ideal 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