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

$
\begin{aligned}
& \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{aligned}
$

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{aligned}
& \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{aligned}
$

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

$
\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|
$

$\Delta_1$ numerator of $x$ is :
$\Delta_1=\left|\begin{array}{lll}d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3\end{array}\right|$
Similarly $\Delta_2=\left|\begin{array}{lll}a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3\end{array}\right|$ and $\Delta_3=\left|\begin{array}{lll}a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3\end{array}\right|$
i) If $\Delta \neq 0$, then the system of equations has a unique finite solution and so equations are consistent, and solutions are $\mathrm{x}=\frac{\Delta_1}{\Delta}, \mathrm{y}=\frac{\Delta_2}{\Delta}, \mathrm{z}=\frac{\Delta_3}{\Delta}$
ii) If $\Delta=0$, and any of $\Delta_1 \neq 0$ or $\Delta_2 \neq 0$ or $\Delta_3 \neq 0$

Then the system of equations is inconsistent and hence no solution exists.
iii) If all $\Delta=\Delta_1=\Delta_2=\Delta_3=0$ 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)