Question

Find AB and BA for the following matrices. A=[abcd]andB=[1001]A = \left[ \begin{matrix} a & b \ c & d \end{matrix} ight] \quad 	ext{and} \quad B = \left[ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} ight]
 Matrix B acts as the multiplicative element for 2 ×	imes 2 square matrices.

Accepted Answer

First, identify the matrices A and B explicitly. Since the problem statement does not provide the actual entries of matrices A and B, you need to write down or obtain the elements of both matrices. Each should be a 2x2 matrix, for example, $$ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} $$ and $$ B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} . To $$find the product $$ AB $$, multiply matrix A by matrix B using the rule for matrix multiplication: the element in the $$ i^{th} $$ row and $$ j^{th} $$ column of $$ AB  is $$calculated as $$ (AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} . $$Perform this for all four elements to get the resulting 2x2 matrix. Similarly, to find the product $$ BA $$, multiply matrix B by matrix A using the same matrix multiplication rule: $$ (BA)_{ij} = b_{i1}a_{1j} + b_{i2}a_{2j} . $$Calculate each element to form the resulting 2x2 matrix. Remember that matrix multiplication is generally not commutative, so $$ AB $$ and $$ BA $$ may not be the same matrix. Carefully compute each product separately. After computing both products, write down the resulting matrices $$ AB $$ and $$ BA $$ explicitly, showing all intermediate multiplication and addition steps for clarity.

Find AB and BA for the following matrices.
$A = [\begin{matrix} a & b \\ c & d \end{matrix}]$
Matrix B acts as the multiplicative element for 2 $\times$ 2 square matrices.

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Key Concepts

Matrix Multiplication

Properties of 2x2 Matrices

Multiplicative Identity Matrix