Find AB and BA for the following matrices. A= and B = Matrix B acts as the multiplicative element for 2 * 2 square matrices.

Verified step by step guidance

1. First, identify the matrices A and B explicitly. Since the problem statement does not provide the entries of matrices A and B, ensure you have the correct 2x2 matrices before proceeding.
2. Recall that matrix multiplication is defined such that for two 2x2 matrices A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} and B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}, the product AB is another 2x2 matrix where each element is calculated by taking the dot product of the rows of A with the columns of B.
3. Calculate the product AB by computing each element as follows: \[(AB)_{11} = a_{11} \times b_{11} + a_{12} \times b_{21}\] \[(AB)_{12} = a_{11} \times b_{12} + a_{12} \times b_{22}\] \[(AB)_{21} = a_{21} \times b_{11} + a_{22} \times b_{21}\] \[(AB)_{22} = a_{21} \times b_{12} + a_{22} \times b_{22}\]
4. Similarly, calculate the product BA by switching the order of multiplication and using the same method: multiply rows of B by columns of A to find each element of BA.
5. After computing both AB and BA, compare the two resulting matrices to observe that matrix multiplication is generally not commutative, meaning AB may not equal BA.

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices by taking the dot product of rows from the first matrix with columns from the second. For two 2x2 matrices, each element of the product is calculated by summing the products of corresponding entries. This operation is not commutative, meaning AB does not necessarily equal BA.

Properties of 2x2 Matrices

A 2x2 matrix has four elements arranged in two rows and two columns. Understanding the structure helps in performing operations like multiplication, addition, and finding determinants. The size compatibility (both matrices being 2x2) ensures that multiplication AB and BA are both defined.

Multiplicative Identity in Matrices

The multiplicative identity matrix for 2x2 matrices is the matrix with ones on the diagonal and zeros elsewhere. Multiplying any 2x2 matrix by this identity matrix leaves it unchanged. Recognizing this helps in understanding the role of matrix B as a multiplicative element and its effect on matrix A.