We need to identify the correct way to solve the matrix equation:
[tex]AX=B[/tex]
First, notice that the order in which we multiply matrices matters. Then, if we multiply one side of the equation by a matrix M from the left, we also need to multiply the other side by M from the left:
[tex]\begin{gathered} MAX=MB \\ \\ \text{ but }MAX\text{ is not necessarily equal to }BM \end{gathered}[/tex]
Also, when we multiply a matrix by its inverse, we obtain the identity matrix:
[tex]\begin{gathered} A^{-1}A=I \\ \\ \Rightarrow A^{-1}AX=IX=X \end{gathered}[/tex]
Then, multiplying both sides of the given equation by A⁻¹, we obtain:
[tex]\begin{gathered} A^{-1}AX=A^{-1}B \\ \\ X=A^{-1}B \end{gathered}[/tex]
Therefore, the answer is:
[tex]X=A^{-1}B[/tex]
