Determine the relationship between matrices A, X and B:
Exemple 4
Considérons les matrices suivantes :
$A=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right],X=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]$ et $B=\left[\begin{array}{l}{b}_1\\ b_2\\ b_3\end{array}\right]$ vérifiant l'équation $AX=B$.

Step 1 ：We are given the matrix equation $AX=B$, where $A=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, $X=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]$ and $B=\left[\begin{array}{l}{b}_1\\ b_2\\ b_3\end{array}\right]$. We are asked to determine the relationship between these matrices.

Step 2 ：The matrix equation $AX=B$ can be solved for $X$ by multiplying both sides of the equation by the inverse of $A$, if it exists. The inverse of a matrix $A$ is denoted $A^{-1}$ and it has the property that $A{A}^{-1}=A^{-1}A=I$, where $I$ is the identity matrix.

Step 3 ：So, if we multiply both sides of the equation by $A^{-1}$, we get $A^{-1}AX=A^{-1}B$, which simplifies to $X=A^{-1}B$. Therefore, to solve the equation, we need to find the inverse of $A$ and then multiply it by $B$.

Step 4 ：However, finding the inverse of a 3x3 matrix is a bit more complex than a 2x2 matrix. The formula for the inverse of a 3x3 matrix is $A^{-1}=\frac{1}{det(A)}adj(A)$, where $det(A)$ is the determinant of $A$ and $adj(A)$ is the adjugate of $A$.

Step 5 ：Once we find the inverse of $A$, we can multiply it by $B$ to get $X$.

Step 6 ：Therefore, the relationship between $A$, $X$, and $B$ is that $X$ is the product of the inverse of $A$ and $B$, or $X=A^{-1}B$.