matrix addition question on adding the unknown terms

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March 20th 2009, 02:44 PM
Craka

matrix addition question on adding the unknown terms

Okay this may sound really like a stupid question, but 'life is like a box of chocolate you never know what your going to get' lol.

Question is when asked to put linear equations into matrix form and than add the matrices together are you suppose to not add the unknown matrix back on it self ie for instance the x1, x2, ...xn matrix.

For instance with the following sets of equations

$ \begin{array}{l} \left\{ \begin{array}{l} 4x_1 + 7x_2 = 2 \\ x_1 + 2x_2 - x_3 = 3 \\ \end{array} \right. \\ \left\{ \begin{array}{l} 7x_2 + 8x_3 = 0 \\ x_1 - x_2 = 17 \\ \end{array} \right. \\ \end{array} $

So my matrices for the above equations are respectively as follows

$ \left[ {\begin{array}{*{20}c} 4 & 7 & 0 \\ 1 & 2 & { - 3} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 2 \\ 3 \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array}} \right] $

$ \left[ {\begin{array}{*{20}c} 0 & 7 & 8 \\ 1 & { - 1} & 0 \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 0 \\ {17} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array}} \right] $

And here is where my question, regarding to the column matrix containing the x terms.

$ \left[ {\begin{array}{*{20}c} 4 & {14} & 8 \\ 2 & 1 & { - 3} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 2 \\ {20} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 + x_1 } \\ {x_2 + x_2 } \\ {x_3 + x_3 } \\ \end{array}} \right] $

Should it be as above or should you not add the x1+x1 , x2+x2 ... etc. If not why ?
March 20th 2009, 06:11 PM
mr fantastic

Quote:

Originally Posted by Craka

Okay this may sound really like a stupid question, but 'life is like a box of chocolate you never know what your going to get' lol.

Question is when asked to put linear equations into matrix form and than add the matrices together are you suppose to not add the unknown matrix back on it self ie for instance the x1, x2, ...xn matrix.

For instance with the following sets of equations

$ \begin{array}{l} \left\{ \begin{array}{l} 4x_1 + 7x_2 = 2 \\ x_1 + 2x_2 - x_3 = 3 \\ \end{array} \right. \\ \left\{ \begin{array}{l} 7x_2 + 8x_3 = 0 \\ x_1 - x_2 = 17 \\ \end{array} \right. \\ \end{array} $

So my matrices for the above equations are respectively as follows

$ \left[ {\begin{array}{*{20}c} 4 & 7 & 0 \\ 1 & 2 & { - 3} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 2 \\ 3 \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array}} \right] $

$ \left[ {\begin{array}{*{20}c} 0 & 7 & 8 \\ 1 & { - 1} & 0 \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 0 \\ {17} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array}} \right] $

And here is where my question, regarding to the column matrix containing the x terms.

$ \left[ {\begin{array}{*{20}c} 4 & {14} & 8 \\ 2 & 1 & { - 3} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} 2 \\ {20} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} {x_1 + x_1 } \\ {x_2 + x_2 } \\ {x_3 + x_3 } \\ \end{array}} \right] $

Should it be as above or should you not add the x1+x1 , x2+x2 ... etc. If not why ?

The correct matrix equation for the equations you've posted is $ \left( \begin{array}{ccc} 4 & 7 & 0 \\ 1 & 2 & -1 \\ 0 & 7 & 8 \\ 1 & -1 & 0 \end{array} \right) $ $ \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) = $ $ \left( \begin{array}{c} 2 \\ 3 \\ 0 \\ 17\end{array} \right) $
March 20th 2009, 08:42 PM
Craka

oh, I think I see. So I leave the unknowns alone?