Cramer’s Rule - Practice Questions & MCQ

Cramer’s law

For the system of equations in two variables:

Let $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$, where

$\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}\ne \frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}$
On solving this equation by cross multiplication, we get

$\begin{array}{rl}& \frac{x}{b_2{c}_1-b_1{c}_2}=\frac{y}{a_1{c}_2-a_2{c}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}\\ & \text{ or }\frac{\mathrm{x}}{\left|\begin{array}{ll}{c}_1& b_1\\ c_2& b_2\end{array}\right|}=\frac{\mathrm{y}}{\left|\begin{array}{ll}{a}_1& c_1\\ a_2& c_2\end{array}\right|}=\frac{1}{\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|}\\ & \text{ or }\mathrm{x}=\frac{\left|\begin{array}{ll}{c}_1& b_1\\ c_2& b_2\end{array}\right|}{\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|},\mathrm{y}=\frac{\left|\begin{array}{ll}{a}_1& c_1\\ a_2& c_2\end{array}\right|}{\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|}\end{array}$

We can observe that the first column in the numerator of x is of constants and 2nd column in the numerator of y is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variables where the third column in the numerator of the value of z will be constant and the denominator will be formed by the value of coefficients of the variables.

For the system of equations in three variables:

Let us consider the system of equations

$\begin{array}{rl}& {\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1\mathrm{z}={\mathrm{d}}_1\\ & {\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2\mathrm{z}={\mathrm{d}}_2\\ & {\mathrm{a}}_3\mathrm{x}+{\mathrm{b}}_3\mathrm{y}+{\mathrm{c}}_3\mathrm{z}={\mathrm{d}}_3\end{array}$

then $\mathrm{\Delta }$, which will be determinant of coefficient of variables, will be

$\mathrm{\Delta }=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

${\mathrm{\Delta }}_1$ numerator of $x$ is :
${\mathrm{\Delta }}_1=\left|\begin{array}{lll}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|$
Similarly ${\mathrm{\Delta }}_2=\left|\begin{array}{lll}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|$ and ${\mathrm{\Delta }}_3=\left|\begin{array}{lll}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|$
i) If $\mathrm{\Delta }\ne 0$, then the system of equations has a unique finite solution and so equations are consistent, and solutions are $\mathrm{x}=\frac{{\mathrm{\Delta }}_1}{\mathrm{\Delta }},\mathrm{y}=\frac{{\mathrm{\Delta }}_2}{\mathrm{\Delta }},\mathrm{z}=\frac{{\mathrm{\Delta }}_3}{\mathrm{\Delta }}$
ii) If $\mathrm{\Delta }=0$, and any of ${\mathrm{\Delta }}_1\ne 0$ or ${\mathrm{\Delta }}_2\ne 0$ or ${\mathrm{\Delta }}_3\ne 0$

Then the system of equations is inconsistent and hence no solution exists.
iii) If all $\mathrm{\Delta }={\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\mathrm{\Delta }}_3=0$ then

System of equations is consistent and it has an infinite number of solutions (except when all three equations represent parallel planes, in which case there is no solution)