Application of Determinant to Systems: Cramer's Rule

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)

A X = B

where A is the matrix coefficient, B the nonhomogeneous term, and X the unknown column-matrix. We have:

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:

\begin{displaymath}x_i = \frac{\det(A_i)}{detA}\;,\;\; \mbox{for $i=1,\cdots,n$}\end{displaymath}

where xi are the unknowns of the system or the entries of X, and the matrix Ai is obtained from A by replacing the ith column by the column B. In other words, we have

\begin{displaymath}x_i = \frac{b_1 A_{1i} + b_2 A_{2i} + \cdots + b_n A_{ni}}{\det(A)}\end{displaymath}

where the bi are the entries of B.

In particular, if the linear system AX = B is homogeneous, meaning $B = {\cal O}$, then if A is invertible, the only solution is the trivial one, that is $X = {\cal O}$. 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 $\det (A) = 0$. This is an important result.

Example. Solve the linear system

\begin{displaymath}\left(\begin{array}{rrr}
1&2&0\\
-1&1&1\\
1&2&3\\
\end{arr...
...ht) = \left(\begin{array}{r}
0\\
1\\
0\\
\end{array}\right).\end{displaymath}

Answer. First note that

\begin{displaymath}\left\vert\begin{array}{rrr}
1&2&0\\
-1&1&1\\
1&2&3\\
\end...
...ert\begin{array}{rrr}
-1&1\\
1&3\\
\end{array}\right\vert = 9\end{displaymath}

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

\begin{displaymath}x = \frac{1}{9} \left\vert\begin{array}{rrr}
0&2&0\\
1&1&1\\...
...array}{rrr}
1&2&0\\
-1&1&1\\
1&2&0\\
\end{array}\right\vert.\end{displaymath}

We leave the details to the reader to find

\begin{displaymath}x = \frac{-4}{9},\; y = \frac{3}{9} = \frac{1}{3},\;\mbox{and}\; z = 0.\end{displaymath}

Note that it is easy to see that z=0. Indeed, the determinant which gives z has two identical rows (the first and the last). We do encourage you to check that the values found for x, y, and z are indeed the solution to the given system.

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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