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. (A+B)(A+B)=A^2+2AB+B^2

Question
Matrices
asked 2020-11-26
Explain why the formula is not valid for matrices. Illustrate your argument with examples.
\((A+B)(A+B)=A^2+2AB+B^2\)

Answers (1)

2020-11-27
Step 1
Consider the provided question,
According to you we have to solve only 38 question.
(38)
Show that why the formula \((A+B)(A+B)=A^2+2AB+B^2\) is not valid for matrices. In the case of matrices,
\((A+B)(A+B)=AA+AB+BA+BB\)
\(= A^2+AB+BA+B^2\)
Since, we know that hhe relation AB=BA is not always correct.
Step 2
Now, consider an example.
Let \(A=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(AB=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix}\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(=\begin{bmatrix}1\cdot0+2\cdot1 & 1(-1)+2\cdot2 \\ 0\cdot0+3\cdot1 & 0(-1)+3\cdot2 \end{bmatrix}\)
\(=\begin{bmatrix}2& 3 \\3 & 6 \end{bmatrix}\)
Step 3
\(A=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(BA=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix}\)
\(=\begin{bmatrix}0\cdot1+(-1)0 & 0\cdot2+(-1)3 \\1\cdot1+2\cdot0 & 1\cdot2+2\cdot3 \end{bmatrix}\)
\(=\begin{bmatrix}0 & -3 \\1 & 8 \end{bmatrix}\)
Here, \(AB \neq BA\)
and as a result \(AB+BA \neq 2AB\)
Thus, the formula \((A+B)(A+B)=A^2+2AB+B^2\) is not correct for all matrices.
0

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