# 3x3 matrix

I have a doubt...

Look this matrix equation:
$$\begin{bmatrix} A\\ B \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix}$$
$$\begin{bmatrix} X\\ Y \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} A\\ B \end{bmatrix}$$
By analogy, should exist a matrix 3x3 such that:
$$\begin{bmatrix} A\\ B\\ C\\ \end{bmatrix} = \begin{bmatrix} ? & ? & ?\\ ? & ? & ?\\ ? & ? & ?\\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix}$$
$$\begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix} = \begin{bmatrix} ? & ? & ?\\ ? & ? & ?\\ ? & ? & ?\\ \end{bmatrix} \begin{bmatrix} A\\ B\\ C\\ \end{bmatrix}$$
So, what values need be replaced in ? for the matrix equation above be right?

I think that my doubt is related with these wikipages:
https://en.wikipedia.org/wiki/Cubic_function#Lagrange.27s_method
https://en.wikipedia.org/wiki/Quartic_function#Solving_by_Lagrange_resolvent

EDIT: The inverse of the 3x3 matrix need be equal to itself.

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Mark44
Mentor
I have a doubt...

Look this matrix equation:
$$\begin{bmatrix} A\\ B \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix}$$
$$\begin{bmatrix} X\\ Y \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} A\\ B \end{bmatrix}$$
Your question is not very complicated. If ##\vec{x} = A\vec{b}##, and A is an invertible matrix, then ##\vec{b} = A^{-1}\vec{x}##. Note that capital letters are usually used for matrix names, and lower case letters are used for vectors or the components of vectors.
Bruno Tolentino said:
By analogy, should exist a matrix 3x3 such that:
$$\begin{bmatrix} A\\ B\\ C\\ \end{bmatrix} = \begin{bmatrix} ? & ? & ?\\ ? & ? & ?\\ ? & ? & ?\\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix}$$
$$\begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix} = \begin{bmatrix} ? & ? & ?\\ ? & ? & ?\\ ? & ? & ?\\ \end{bmatrix} \begin{bmatrix} A\\ B\\ C\\ \end{bmatrix}$$
Yes, as long as the matrix has an inverse. The equation I wrote is applicable for any square matrix A that has an inverse.
Bruno Tolentino said:
None of these links is helpful, as far as I can see. Any linear algebra textbook will have a section on finding the inverse of a square matrix.
Bruno Tolentino said:
EDIT: The inverse of the 3x3 matrix need be equal to itself.
Of course.

Last edited:
mathman
The problem in matrix form is find A (if it exist other than A=I) where $A^2=I$. In scalar form it is 9 equations with 9 unknowns.