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 $A X=B$.

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 \(AA^{-1} = A^{-1}A = I\), 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 \(X = A^{-1}B\).