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I know that a matrix A given by: $$ \begin{vmatrix} (a + x) & (b + y) & (c + z) \\\ p & q & r \\\ s & t & u \end{vmatrix} $$ can be split as $$ \begin{vmatrix} a & b & c \\\ p & q & r \\\ s & t & u \end{vmatrix} + \begin{vmatrix} x & y & z \\\ p & q & r \\\ s&t&u \end{vmatrix} $$.

Can we do something like this? If a matrix B is given by: $$ \begin{vmatrix} a + x & b + y & c + z \\\ p + d & q + e & r + f \\\ s + h & t + i & u + j \end{vmatrix} $$ Can we split it into:

$$ \begin{vmatrix} a & b & c \\\ p & q & r \\\ s & t & u \end{vmatrix} + \begin{vmatrix} x & y & z \\\ d & e & f \\\ s&t&u \end{vmatrix} $$.

WorldGov
  • 947

1 Answers1

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The answer to your question is that it is possible, but its not as simple as the formula you proposed.

Here is the correct calculation :

$$ \begin{vmatrix} a + x & b + y & c + z \\\ p + d & q + e & r + f \\\ s + h & t + i & u + j \end{vmatrix} $$

$$= \begin{vmatrix} x & y & z \\\ p + d & q + e & r + f \\\ s + h & t + i & u + j \end{vmatrix} + \begin{vmatrix} a & b & c \\\ p + d & q + e & r + f \\\ s + h & t + i & u + j \end{vmatrix} $$

$$= \begin{vmatrix} x & y & z \\\ d & e & f \\\ s + h & t + i & u + j \end{vmatrix} + \begin{vmatrix} x & y & z \\\ p & q & r \\\ s + h & t + i & u + j \end{vmatrix} + \begin{vmatrix} a & b & c \\\ d & e & f \\\ s + h & t + i & u + j \end{vmatrix} + \begin{vmatrix} a & b & c \\\ p & q & r \\\ s + h & t + i & u + j \end{vmatrix} $$

$$= \begin{vmatrix} x & y & z \\\ d & e & f \\\ s & t & u \end{vmatrix} + \begin{vmatrix} x & y & z \\\ d & e & f \\\ h & i & j \end{vmatrix} + \begin{vmatrix} x & y & z \\\ p & q & r \\\ s & t & u \end{vmatrix} + \begin{vmatrix} x & y & z \\\ p & q & r \\\ h & i & j \end{vmatrix} + \begin{vmatrix} a & b & c \\\ d & e & f \\\ s & t & u \end{vmatrix} + \begin{vmatrix} a & b & c \\\ d & e & f \\\ h & i & j \end{vmatrix} + \begin{vmatrix} a & b & c \\\ p & q & r \\\ s & t & u \end{vmatrix} + \begin{vmatrix} a & b & c \\\ p & q & r \\\ h & i & j \end{vmatrix} $$

Spencer
  • 12,271