How can a nonsingular matrix be used to solve a matrix equation?

A square matrix is considered nonsingular if its determinant is not equal to zero. For such a matrix, there exists an inverse matrix, denoted as $A^{-1}$, which satisfies the condition $A{A}^{-1}=A^{-1}A=I$, where $I$ is the identity matrix.12

To solve a matrix equation of the form $AX=B$ (where $A$ and $B$ are known matrices and $X$ is the unknown), you can utilize the nonsingular property of $A$ to isolate $X$:

1. Multiply both sides of the equation on the left by the inverse matrix $A^{-1}$:
   $A^{-1}(AX)=A^{-1}B$
2. Apply the associative law of matrix multiplication:
   $(A^{-1}A)X=A^{-1}B$
3. Since $A^{-1}A=I$ and $IX=X$, the equation simplifies to:
   $X=A^{-1}B$

This method provides a unique solution for the unknown matrix $X$.1