A Simple Formula to Calculate the Sum of Elements of the Product of Second-Order Matrices
Formulas
The sum of the elements of the quadratic matrix $\begin{bmatrix} { a }&{ b } \\ { c }&{ d } \end{bmatrix} \begin{bmatrix} { p }&{ q } \\ { r }&{ s } \end{bmatrix}$ is as follows. $$ {(a+c)(p+q)}+{(b+d)(r+s)} $$
Description
You may have encountered many problems asking to find the sum of the elements of the product of two quadratic matrices. As everyone knows, though multiplying matrices is not hard, it does take time and is quite bothersome. Therefore, we introduce a formula that drastically reduces the computational effort.
Derivation
$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} { ap+br }&{ aq+bs } \\ { cp+dr }&{ cq+ds } \end{bmatrix} $$ Thus, the sum of the elements of this matrix is $$ \begin{align*} & (ap+br)+(aq+bs)+(cp+dr)+(cq+ds) \\ =& (ap+aq+cp+cq)+(br+bs+dr+ds) \\ =& (a+c)(p+q)+(b+d)(r+s) \end{align*} $$
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