Math Problem Statement

Find AB and BA for matrices A = [[2, 3], [1, 4]] and B = [[5, 1], [0, 3]].

Solution

The image contains two matrices, AA and BB, and a task asking to find the products ABAB and BABA. The matrices are given as:

A=[2314],B=[5103]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 1 \\ 0 & 3 \end{bmatrix}

Step 1: Matrix Multiplication ABAB

Matrix multiplication follows the rule that the element at position (i,j)(i, j) in the product is the dot product of the ii-th row of the first matrix with the jj-th column of the second matrix. Let's calculate ABAB.

2 \cdot 5 + 3 \cdot 0 & 2 \cdot 1 + 3 \cdot 3 \\ 1 \cdot 5 + 4 \cdot 0 & 1 \cdot 1 + 4 \cdot 3 \end{bmatrix} = \begin{bmatrix} 10 & 11 \\ 5 & 13 \end{bmatrix}$$ ### Step 2: Matrix Multiplication $$BA$$ Now, let's calculate $$BA$$. $$BA = \begin{bmatrix} 5 \cdot 2 + 1 \cdot 1 & 5 \cdot 3 + 1 \cdot 4 \\ 0 \cdot 2 + 3 \cdot 1 & 0 \cdot 3 + 3 \cdot 4 \end{bmatrix} = \begin{bmatrix} 11 & 19 \\ 3 & 12 \end{bmatrix}$$ ### Final Results: $$AB = \begin{bmatrix} 10 & 11 \\ 5 & 13 \end{bmatrix}, \quad BA = \begin{bmatrix} 11 & 19 \\ 3 & 12 \end{bmatrix}$$ Let me know if you'd like further details or have any questions. Here are 5 related questions to expand on this: 1. How do you determine when matrix multiplication is possible? 2. Can matrix multiplication be commutative? Why or why not? 3. What are the properties of matrix multiplication in terms of associativity and distributivity? 4. How can we compute the determinant of the matrices $$A$$ and $$B$$? 5. What happens when you multiply a matrix by the identity matrix? ### Tip: Matrix multiplication is not always commutative, meaning $$AB \neq BA$$, as shown in this example. Always check the dimensions carefully before multiplying!

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Linear Algebra

Formulas

For AB: (i, j)-th element = row i of A · column j of B
For BA: (i, j)-th element = row i of B · column j of A

Theorems

Matrix multiplication is associative but not commutative.
Matrix dimensions must be compatible for multiplication.

Suitable Grade Level

Grades 9-12 or early university level