Elementary Row Operations

What I'm wondering is: "If multiplying the equation by the inverse (which is found via the determinant) gives you the solutions (for the 3 variables), then what are elementary row operatons for?".

For a two variable equation in a system with two equations (which could also be solved by substitution):

\(\displaystyle A = xy\)

\(\displaystyle A(A^{-1})x = y(A^{-1})\)

so that leads to (because the A on the left is canceled out cause it's multiplied by it's inverse)

\(\displaystyle x= y(A^{-1}) =\) The two solutions to the problem.

http://www.stattrek.com/matrix-algebra/system-of-equations.aspx

You could use the same equation above to solve for 3 unknown variables. However, you wouldn't be able to use substitution or elimination in that case.
 
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Jason76 said:
What I'm wondering is: "If multiplying the equation by the inverse (which is found via the determinant) gives you the solutions (for the 3 variables), then what are elementary row operatons for?".

For a two variable equation in a system with two equations (which could also be solved by substitution):

\(\displaystyle A = xy\)

\(\displaystyle A(A^{-1})x = y(A^{-1})\)

so that leads to (because the A on the left is canceled out cause it's multiplied by it's inverse)

\(\displaystyle x= y(A^{-1}) =\) The two solutions to the problem.

http://www.stattrek.com/matrix-algebra/system-of-equations.aspx

You could use the same equation above to solve for 3 unknown variables. However, you wouldn't be able to use substitution or elimination in that case.
Click to expand...

In real world problems, the matrix inversion technique (using traditional methods) becomes computationally inefficient due to enormous size of the matrix. Most of the time, special alogorithms are used (for solution) due to "sparse" nature of the matrix.
 
Subhotosh Khan said:
In real world problems, the matrix inversion technique (using traditional methods) becomes computationally inefficient due to enormous size of the matrix. Most of the time, special alogorithms are used (for solution) due to "sparse" nature of the matrix.
Click to expand...

Here is an easier way, as you said:

http://mathonweb.com/help/backgd3d.htm

It transforms the equation into the "triangular echelon form", and then does substitution to find the variables.

What are some other ways to get solutions?
 
Last edited by a moderator:
Jason76 said:
What I'm wondering is: "If multiplying the equation by the inverse (which is found via the determinant) gives you the solutions (for the 3 variables), then what are elementary row operatons for?".

For a two variable equation in a system with two equations (which could also be solved by substitution):

\(\displaystyle A = xy\)

\(\displaystyle A(A^{-1})x = y(A^{-1})\)

so that leads to (because the A on the left is canceled out cause it's multiplied by it's inverse)

\(\displaystyle x= y(A^{-1}) =\) The two solutions to the problem.

http://www.stattrek.com/matrix-algebra/system-of-equations.aspx

You could use the same equation above to solve for 3 unknown variables. However, you wouldn't be able to use substitution or elimination in that case.
Click to expand...

Jason,
I love your question, and the reason is simple:
Matrix solutions using row operations are often easier to do, especially if one does not remember how to calculate the inverse using the determinant.
The row operations are easy to remember -- they are each legal matrix multiplies.

I encourage you to discover a matrix to multiply by (on the right) to achieve the row operation that you apply.
 
Subhotosh Khan said:
What calculations are you talking about? They are simply finding the determinants of the corresponding matrices.
Click to expand...

Ok, what is the formula for finding a determinant for a 3X3 matrix? How is it applied to at least one of the (3, 0, 0) matrices?

Jason,
I love your question, and the reason is simple:
Matrix solutions using row operations are often easier to do, especially if one does not remember how to calculate the inverse using the determinant.
The row operations are easy to remember -- they are each legal matrix multiplies.

I encourage you to discover a matrix to multiply by (on the right) to achieve the row operation that you apply.
Click to expand...

Probably because determinant formulas might be more difficult to remember, but maybe not.
 
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It's surely not that difficult to remember how to find a determinant of a three by three matrix
1) expand by minors on a row or column.
2) sum of products of the "upper left to lower right" diagonals minus sum of products of "upper right to lower left" diagonals.
or, my favorite
3) use the row operation "add a multiple of one row to another" to reduce to a triangular matrix, then multiply the numbers on the main diagonal.

For example, to find the determinant
\(\displaystyle \left|\begin{array}{ccc}1 & 3 & 2 \\ 2 & 1 & 4 \\ 4 & 1 & 3\end{array}\right|\)

1) Expand by the top row:
\(\displaystyle \left|\begin{array}{cc}1 & 4 \\ 1 & 3\end{array}\right|- 3\left|\begin{array}{cc}2 & 4 \\ 4 & 3\end{array}\right|+ 2\left|\begin{array}{cc}2 & 1 \\ 4 & 1\end{array}\right|\)\(\displaystyle = (3- 4)- 3(6- 16)+ 2(2- 4)= -1+ 30- 4= 25\).

2) Write as
\(\displaystyle \left|\begin{array}{ccc}1 & 3 & 2 \\ 2 & 1 & 4 \\ 4 & 1 & 3\end{array}\right|\left|\begin{array}{cc}1 & 3\\ 2 & 1 \\ 4 & 1 \end{array}\right|\)
so that the "sum of products of the 'upper left to lower right' diagonals" is (1)(1)(3)+ (3)(4)(4)+ (2)(2)(1)= 3+ 48+ 4= 55, the "sum of products of 'upper right to lower left diagonals' is 2(1)(4)+ 1(4)(1)+ (3)(2)(3)= 8+ 4+ 18= 30 and their difference is 25.

3) Subtract twice the first row from the second and subtract four times the first row from the third to get
\(\displaystyle \left|\begin{array}{ccc}1 & 3 & 2 \\ 0 & -5 & 0 \\ 0 & -11 & -5 \end{array}\right|\)
Then add 11/5 times the second row to the third to get
\(\displaystyle \left|\begin{array}{ccc}1 & 3 & 2 \\ 0 & -5 & 0 \\ 0 & 0 & -5\end{array}\right|\)
so that the determinant is (1)(-5)(-5)= 25.
 
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uTube: 01-3 Elementary row operations (part 1/2)
is the first of 3 VERY GOOD videos, describing use of row operations.
It will help you see why it is often easier, especially if the coefficients are integers.

By the way it is possible to use row operations to invert a matrix, without determinants.
After you become comfortable with using row operations to solve a system of equations, post a question about inverting a matrix using row operations.

Don't get me wrong, determinants and inverses have their place. There is no judgment about which row operation to use next. That makes programming a computer to solve inverses simpler.
 
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