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:

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

where x_i are the unknowns of the system or the entries of X, and the matrix A_i is obtained from A by replacing the i^th column by the column B. In other words, we have

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

where the b_i 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

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

Answer. First note that

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

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

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

We leave the details to the reader to find

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

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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