Cramer’s Rule

Use Cramer’s Rule to solve each for each of the variables.

• (a)  $\begin{array}{rcrcr}x&-&y&=&4\\ -x&+&2y&=&-7\end{array}$     (b)  $\begin{array}{rcrcr}-2x&+&y&=&-2\\ x&-&2y&=&-2\end{array}$

Use Cramer’s Rule to solve this system for $z$.

Prove Cramer’s Rule.

Here is an alternative proof of Cramer’s Rule that doesn’t overtly contain any geometry. Write $X_{i}$ for the identity matrix with column $i$ replaced by the vector $\vec{x}$ of unknowns $x_{1}$, …, $x_{n}$.

• Observe that $AX_{i}=B_{i}$.

• Take the determinant of both sides.

Suppose that a linear system has as many equations as unknowns, that all of its coefficients and constants are integers, and that its matrix of coefficients has determinant $1$. Prove that the entries in the solution are all integers. (Remark. This is often used to invent linear systems for exercises.)

Use Cramer’s Rule to give a formula for the solution of a two equations/two unknowns linear system.

Can Cramer’s Rule tell the difference between a system with no solutions and one with infinitely many?

The first picture in this Topic (the one that doesn’t use determinants) shows a unique solution case. Produce a similar picture for the case of infinitely many solutions, and the case of no solutions.