# Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n times n matrices A and B. a) (a+b)^2=a^2+2ab+b^2 b) (a+b)(a-b)=a^2-b^2

Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by $n×n$ matrices A and B. $a\right)\left(a+b{\right)}^{2}={a}^{2}+2ab+{b}^{2}$
$b\right)\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

wornoutwomanC

Step 1
(a) Consider the matrices A and B of order n.
Here,
$\left(A+B{\right)}^{2}=\left(A+B\right)\left(A+B\right)$
$\left(A+B{\right)}^{2}=A\left(A+B\right)+B\left(A+B\right)$
$\left(A+B{\right)}^{2}=AA+AB+BA+BB$
$\left(A+B{\right)}^{2}={A}^{2}+AB+BA+{B}^{2}$
As $n×n$ matrices are not always commutative, the formula $\left(A+B{\right)}^{2}={A}^{2}+2AB+{B}^{2}$ will not always work in general for all matrices.
Step 2
(b) Consider the matrices A and B of order n.
Here,
$\left(A+B\right)\left(A-B\right)=A\left(A+B\right)-B\left(A+B\right)$
$\left(A+B\right)\left(A-B\right)=AA+AB-BA-BB$
$\left(A+B\right)\left(A-B\right)={A}^{2}+AB-BA-{B}^{2}$
As $n×n$ matrices are not always commutative, the formula

Jeffrey Jordon