Formulate Cramer's rule for solving systems of linear equations, stating conditions under which the rule is applicable. Prove Cramer's rule for systems of two equations with two unknowns.
So I just wanted to double check my logic for this question, would something along these lines be correct:
Rule – Only applicable for square matrices that have a non zero determinant.
$$a_1x + b_1y = c_1$$
$$a_2x + b_2y = c_2$$
$$x = \frac{D_1}D,y = \frac {D_2}D$$
Where
\begin{align*}
D& = \begin{vmatrix}a_1&b_1\\a_2&b_2\\ \end{vmatrix} &
D_1 &= \begin{vmatrix}c_1&b_1\\c_2&b_2\\ \end{vmatrix} &
D_2 &= \begin{vmatrix}a_1&c_1\\a_2&c_2\\ \end{vmatrix}
\end{align*}
Best Answer
This is the correct formulation for the $2\times 2$ case but you still need to prove that these formulas work. So, to finish you must show that \begin{array}{crcrcrcrcr} a_1\dfrac{D_1}{D} &+& b_1\dfrac{D_2}{D} &=& c_1 \\ a_2\dfrac{D_1}{D} &+& b_2\dfrac{D_2}{D} &=& c_2 \end{array} The first of these two equations is proved by \begin{align*} a_1\dfrac{D_1}{D} + b_1\dfrac{D_2}{D} &= a_1\frac{\begin{vmatrix}c_1&b_1\\ c_2&b_2\end{vmatrix}}{\begin{vmatrix}a_1&b_1\\ a_2&b_2\end{vmatrix}}+b_1\frac{\begin{vmatrix}a_1&c_1\\ a_2&c_2\end{vmatrix}}{\begin{vmatrix}a_1&b_1\\ a_2&b_2\end{vmatrix}} \\ &= a_1\frac{c_1b_2-b_1c_2}{a_1b_2-b_1a_2}+b_1\frac{a_1c_2-c_1a_2}{a_1b_2-b_1a_2} \\ &= \frac{a_1c_1b_2-a_1b_1c_2+b_1a_1c_2-b_1c_1a_2}{a_1b_2-b_1a_2} \\ &= \frac{a_1c_1b_2-b_1c_1a_2}{a_1b_2-b_1a_2} \\ &= c_1\frac{a_1b_2-b_1a_2}{a_1b_2-b_1a_2} \\ &= c_1 \end{align*} Can you prove the second equation?
Cramer's rule in the $n\times n$ case may be stated as follows.
The system $A\vec x=\vec b$ is solved by $$ x_j=\frac{\det A_j}{\det A} $$ where $A_j$ is the $n\times n$ matrix obtained by replacing the $j$th column of $A$ with $\vec b$.