Can I use back substitution to invert a matrix?

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In summary, the conversation discusses calculating a matrix of derivatives and its inverse, particularly for a triangular matrix using the "back substitution" algorithm.
  • #1
NewStudent200
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Hello all,

Say I had n equations, with n variables, so that
C1 = C1(f1), C1 is a funtion of f1 only
C2 = C2(f1, f2), C2 is a function of f1 and f2
...
Cn = Cn(f1, f2...fn), Cn is a function of all n vairables

I can calculate the matrix dC/ df where each line is the derivative of Ci with respect to the n vairables, The matrix will look something lile

a 0 0 0... 0
b c 0 0... 0
...
u v w x... z

Now if I was after the matrix df/dC then am I basically just trying to invert this matrix?

Thanks very much for any help
 
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  • #2
NewStudent200 said:
Hello all,

Say I had n equations, with n variables, so that
C1 = C1(f1), C1 is a funtion of f1 only
C2 = C2(f1, f2), C2 is a function of f1 and f2
...
Cn = Cn(f1, f2...fn), Cn is a function of all n vairables

I can calculate the matrix dC/ df where each line is the derivative of Ci with respect to the n vairables, The matrix will look something lile

a 0 0 0... 0
b c 0 0... 0
...
u v w x... z

Now if I was after the matrix df/dC then am I basically just trying to invert this matrix?

Thanks very much for any help
Yes, and, for a triangular matrix, that is fairly simple.
 
  • #3
Great. Thanks very much. Is there a particular algorithm that is recommended for a triangular matrix?

Regards,
 
  • #4
Just "back substitution". If your matrix is, say,
[tex]\begin{bmatrix}a & 0 & 0 \\ b & c & 0\\ d & e & f\end{bmatrix}[/tex]
then its inverse will be a matrix of the form
[tex]\begin{bmatrix}u & 0 & 0 \\ v & w & 0\\ x & y & z\end{bmatrix}[/tex]
such that
[tex]\begin{bmatrix}a & 0 & 0 \\ b & c & 0\\ d & e & f\end{bmatrix}\begin{bmatrix}u & 0 & 0 \\ v & w & 0\\ x & y & z\end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}[/tex]
You must have au= 1 so u= 1/a. Then bu+ cv= b/a+ cv= 0 so v= -b/ac. cw= 1 so w= 1/c, etc.
 

FAQ: Can I use back substitution to invert a matrix?

What is an inverse matrix?

An inverse matrix is a square matrix that, when multiplied by another square matrix, produces the identity matrix. In other words, it "undoes" the effects of the original matrix.

Why would I want to use an inverse matrix?

Inverse matrices are useful in many areas of mathematics, including solving systems of linear equations, finding determinants, and solving differential equations. They also have applications in physics, engineering, and computer graphics.

How do I find the inverse of a matrix?

The inverse of a matrix can be found using various methods, such as Gaussian elimination, using the adjugate matrix, or using the inverse formula. It is important to note that not all matrices have an inverse.

Can I use an inverse matrix to divide by a matrix?

No, you cannot use an inverse matrix to divide by a matrix. Unlike scalar numbers, matrices do not have a defined division operation. Instead, you can use the inverse matrix to solve a matrix equation, similar to solving for a variable in a regular algebraic equation.

Are there any limitations to using an inverse matrix?

Yes, there are limitations to using an inverse matrix. As mentioned before, not all matrices have an inverse. Additionally, the inverse of a matrix only exists if the matrix is square and has a non-zero determinant. It is also important to be cautious when using inverse matrices in calculations, as rounding errors can lead to inaccurate results.

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