Expert Answer to "Explain why the formula is not valid for..."

Harlen Pritchard

Answered question

2020-11-26

Explain why the formula is not valid for matrices. Illustrate your argument with examples.

(A + B) (A + B) = A^{2} + 2 A B + B^{2}

SchulzD

Skilled2020-11-27Added 83 answers

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} + 2 A B + B^{2}

is not valid for matrices. In the case of matrices,

(A + B) (A + B) = A A + A B + B A + B B

= A^{2} + A B + B A + B^{2}

Since, we know that hhe relation AB=BA is not always correct.
Step 2
Now, consider an example.
Let

A = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] and B = [\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix}]

A B = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] [\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix}]

= [\begin{matrix} 1 \cdot 0 + 2 \cdot 1 & 1 (- 1) + 2 \cdot 2 \\ 0 \cdot 0 + 3 \cdot 1 & 0 (- 1) + 3 \cdot 2 \end{matrix}]

= [\begin{matrix} 2 & 3 \\ 3 & 6 \end{matrix}]

Step 3

A = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] and B = [\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix}]

B A = [\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}]

= [\begin{matrix} 0 \cdot 1 + (- 1) 0 & 0 \cdot 2 + (- 1) 3 \\ 1 \cdot 1 + 2 \cdot 0 & 1 \cdot 2 + 2 \cdot 3 \end{matrix}]

= [\begin{matrix} 0 & - 3 \\ 1 & 8 \end{matrix}]

Here,

A B \neq B A

and as a result

A B + B A \neq 2 A B

Thus, the formula

(A + B) (A + B) = A^{2} + 2 A B + B^{2}

is not correct for all matrices.

Jeffrey Jordon

Expert2022-01-22Added 2605 answers

Answer is given below (on video)

Jeffrey Jordon

Expert2022-08-23Added 2605 answers

Answer is given below (on video)

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