# Explain why the formula is not valid for matrices. Illustrate your argument with examples. (A+B)(A+B)=A^2+2AB+B^2

Explain why the formula is not valid for matrices. Illustrate your argument with examples.
$\left(A+B\right)\left(A+B\right)={A}^{2}+2AB+{B}^{2}$
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SchulzD
Step 1
Consider the provided question,
According to you we have to solve only 38 question.
(38)
Show that why the formula $\left(A+B\right)\left(A+B\right)={A}^{2}+2AB+{B}^{2}$ is not valid for matrices. In the case of matrices,
$\left(A+B\right)\left(A+B\right)=AA+AB+BA+BB$
$={A}^{2}+AB+BA+{B}^{2}$
Since, we know that hhe relation AB=BA is not always correct.
Step 2
Now, consider an example.
Let
$AB=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 2\end{array}\right]$
$=\left[\begin{array}{cc}1\cdot 0+2\cdot 1& 1\left(-1\right)+2\cdot 2\\ 0\cdot 0+3\cdot 1& 0\left(-1\right)+3\cdot 2\end{array}\right]$
$=\left[\begin{array}{cc}2& 3\\ 3& 6\end{array}\right]$
Step 3

$BA=\left[\begin{array}{cc}0& -1\\ 1& 2\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$
$=\left[\begin{array}{cc}0\cdot 1+\left(-1\right)0& 0\cdot 2+\left(-1\right)3\\ 1\cdot 1+2\cdot 0& 1\cdot 2+2\cdot 3\end{array}\right]$
$=\left[\begin{array}{cc}0& -3\\ 1& 8\end{array}\right]$
Here, $AB\ne BA$
and as a result $AB+BA\ne 2AB$
Thus, the formula $\left(A+B\right)\left(A+B\right)={A}^{2}+2AB+{B}^{2}$ is not correct for all matrices.
Jeffrey Jordon