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Compute the following $\;\begin{bmatrix}a^2+b^2 & b^2+c^2\\a^2+c^2 & a^2+b^2\end{bmatrix}+\begin{bmatrix}2ab & 2bc\\-2ac & 2ab\end{bmatrix}$

Note: This is the 2nd part of a 4 part question, which is split as 4 separate questions here.

1 Answer

Toolbox:
  • The sum A+B of two m-by-n matrices A and B is calculated entrywise: $(A+B)_{i,j}=A_{i,j}+B_{i,j}$ where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
$\begin{bmatrix}a^2+b^2 & b^2+c^2\\a^2+c^2 & a^2+b^2\end{bmatrix}+\begin{bmatrix}2ab & 2bc\\-2ac & 2ab\end{bmatrix} = \begin{bmatrix}a^2+b^2+2ab & b^2+c^2+2bc\\a^2+c^2-2ac & a^2+b^2-2ab\end{bmatrix}$
Therefore, $\begin{bmatrix}a^2+b^2 & b^2+c^2\\a^2+c^2 & a^2+b^2\end{bmatrix}+\begin{bmatrix}2ab & 2bc\\-2ac & 2ab\end{bmatrix} = \begin{bmatrix}(a+b)^2 & (b+c)^2\\(a-c)^2 & (a-b)^2\end{bmatrix}$
answered Feb 27, 2013 by balaji.thirumalai
 

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