We have seen that determinant may be useful in finding the inverse of a nonsingular matrix. We can use these findings in solving linear systems for which the matrix coefficient is nonsingular (or invertible).

Consider the linear system (in matrix form)

where

**Theorem.** The linear system *AX* = *B* has a unique solution if and only if *A* is invertible. In this case, the solution is given by the so-called **Cramer's formulas**:

where

where the

In particular, if the linear system *AX* = *B* is homogeneous, meaning
,
then if *A* is invertible, the only solution is the trivial one, that is
.
So if we are looking for a nonzero solution to the system, the matrix coefficient *A* must be singular or noninvertible. We also know that this will happen if and only if
.
This is an important result.

**Example.** Solve the linear system

**Answer.** First note that

which implies that the matrix coefficient is invertible. So we may use the Cramer's formulas. We have

We leave the details to the reader to find

Note that it is easy to see that

**Remark.** Remember that Cramer's formulas are only valid for linear systems with an invertible matrix coefficient.

**
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**Author**: M.A. Khamsi

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