If A is a matrix and |A| ≠ 0, then solution of the equation XA = B is:

The correct option is X = BA^-1.

Let's understand how we arrive at this solution step-by-step.

We are given the matrix equation:
XA = B

We are also given that $\left|A\right|\ne 0$, which means that the matrix A is non-singular. Since A is non-singular, its multiplicative inverse, denoted as A^-1, exists.

To solve for X, we need to isolate it. In matrix algebra, division is not defined, so we must multiply by the matrix inverse. Since matrix multiplication is non-commutative (meaning $PQ\ne QP$ in general), the order of multiplication is extremely important.

In our equation XA = B, the matrix A is multiplied on the right side of X. Therefore, to eliminate A, we must post-multiply (multiply from the right) both sides of the equation by A^-1:

(XA)A^-1 = BA^-1

Using the associative property of matrix multiplication, we can regroup the terms on the left-hand side:
X(AA^-1) = BA^-1

By definition, multiplying a matrix by its inverse yields the identity matrix, I (i.e., $A{A}^{-1}=I$):
XI = BA^-1

Since multiplying any matrix by the identity matrix leaves it unchanged (i.e., $XI=X$), we get:
X = BA^-1