Math Problem Statement
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))
Solution
We are given two matrices:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Determinants
Scalar Multiplication
Formulas
Matrix product: AB = A × B
Determinants are defined for square matrices
Theorems
Non-square matrices do not have determinants
Matrix multiplication is not commutative
Suitable Grade Level
Grades 10-12