Question

The system of equations which can be solved by Cramer’s rule have:
(A) Unique solution
(B) No solution
(C) Infinitely many solutions
(D) Two solutions

Answer 1

Hint: The given question revolves around the concepts of solving multiple linear equations using matrices. A system of linear equations can be solved using matrices with the help of Cramer’s rule. Cramer’s rule is an explicit formula for the solution of a system of linear equations which should be learnt by heart before solving the problem given to us.

Complete answer:
To find the number of solutions of the system of equations that can be solved by Cramer’s rule.
Let us assume three equations in variables x, y and z. Then, we have represent the system of equations in the form of matrices as $AX = B$ where the matrix X is the variable matrix consisting of all the variables, A matrix is the coefficient matrix involving all the coefficients of variables in the equations and matrix B is the constant matrix involving all the constant terms in the equations.
So, Cramer’s rule involves finding the values of variables by replacing a column of coefficients in the coefficient matrix by the constant vector and then dividing the determinant of the resultant matrix by the determinant of the coefficient matrix.
Hence, the values of variables x, y and z in our example can be found as: $x = \dfrac{{\left| {{A_x}} \right|}}{{\left| A \right|}}$, $y = \dfrac{{\left| {{A_y}} \right|}}{{\left| A \right|}}$ and \[z = \dfrac{{\left| {{A_z}} \right|}}{{\left| A \right|}}\].
Now, we can see that there are some limitations in solving a system of linear equations by using the Cramer’s rule. The values of variables cannot be found using Cramer's rule if the value of the determinant of the coefficient matrix is zero. The determinant value is zero if any two rows or columns are exactly the same.
So, this can happen only when two of the equations have the exact same coefficients. Hence, the system of equations that can be solved with the help of Cramer’s rule cannot have no solution as well as infinitely many solutions. Conversely, the equations which can be solved by Cramer’s rule have a unique solution. Hence, option (A) is the correct option.

Note:
There are many other ways of finding the solutions to the system of equations apart from using the Cramer’s rule. Cramer’s rule may be an uncomfortable method to find the solution to the system of equations as it involves solving the tedious determinants. One must know the concepts of matrices and method of solving the system of equations using matrices before attempting such questions as it requires a clear understanding of the theory.