# [SOLVED] Elementary Matrices

• Feb 22nd 2009, 08:19 AM
Incanus
[SOLVED] Elementary Matrices
Which two elementary operations transform the following matrix:
Code:

    2    -6    3    -3    -6     4    -2    -6    -6    0     -1    -3    -9    6    4     7    3    3    -9    2
Into this one:
Code:

    2    -6    3    -3    -6     4    -2    -6    -6    0   -27    99  -18    27    78     7    3    3    -9    2
Thanks for any help!
• Feb 22nd 2009, 09:55 AM
HallsofIvy
Quote:

Originally Posted by Incanus
Which two elementary operations transform the following matrix:
Code:

    2    -6    3    -3    -6     4    -2    -6    -6    0     -1    -3    -9    6    4     7    3    3    -9    2
Into this one:
Code:

    2    -6    3    -3    -6     4    -2    -6    -6    0   -27    99  -18    27    78     7    3    3    -9    2
Thanks for any help!

What exactly your difficulty? I see that only the third row has changed so whatever elementary operations (row operations) are done must be done to it. There are three types of row operations: swap two rows (obviously not here), multiply one row by number, add a multiple of one row to another.

You might try this: if we were to multiply the row by a, then add b times the first row, the first number would be -a+ 2b= -27 and the second number would be -3a- 6b= 99. That gives two equations to solve for a and b. Now try the third number: For those values of a and b, is -9a+ 3b= -18? If yes, check the other numbers but you are probably done. If not try b times the second row: -a+ 4b= -27, -3a-2b= 99, etc.
If none of those work, then it must be that "add a multiple of one row to the third row" was used twice: Try a times the first row pluls b times the second row added to the third: 2a+ 4b- 1= -27, 2a- 2b- 3= 99, again two equations in two unknowns that can be checked by the other numbers in the third row.
Tedious, but just algebra.
• Feb 22nd 2009, 02:00 PM
Soroban
Hello, Incanus!

Never seen a guessing game with matrices.
Wonder what the purpose of the problem is?

Quote:

Which two elementary operations transform

this matrix . $\begin{pmatrix}2 &\text{-}6&3&\text{-}3&\text{-}6\\ 4&\text{-}2&\text{-}6&\text{-}6&0\\ \text{-}1&\text{-}3&\text{-}9&6&4\\ 7&3&3&\text{-}9&2\end{pmatrix}$ . into this one: . $\begin{pmatrix}2&\text{-}6&3&\text{-}3&\text{-}6\\ 4&\text{-}2&\text{-}6&\text{-}6&0\\ \text{-}27 &99&\text{-}18&27&78\\ 7&3&3&\text{-}9&2\end{pmatrix}$

. . $\begin{array}{c}\text{Multiply }R_3\text{ by -3.} \\ \\[-4mm]
\text{Multiply }R_1\text{ by -15.} \\ \\[-4mm]