Cramer’s Rule is applicable only when:
A
The system is inconsistent
B
The coefficient matrix is singular
C
The determinant of coefficient matrix is non-zero
D
The system is homogeneous
Analysis & Theory
Cramer’s Rule applies if det(A) ≠ 0, i.e., the coefficient matrix is non-singular.
In Cramer’s Rule, the solution for variable x in a 3-variable system is given by:
Analysis & Theory
x = det(A₁)/det(A), where A₁ is obtained by replacing the first column of A with the constant column.
If det(A) = 0, then:
A
The system has a unique solution
B
Cramer’s Rule cannot be applied
C
The system always has infinitely many solutions
D
The system is always inconsistent
Analysis & Theory
Cramer’s Rule requires det(A) ≠ 0. If det(A) = 0, the system may have no solution or infinitely many solutions.
For a system of n equations in n unknowns, how many determinants need to be calculated in Cramer’s Rule?
Analysis & Theory
We need det(A) and det(A₁), det(A₂), …, det(Aₙ). Total = n + 1 determinants.
In a system of 2 equations in 2 variables, if det(A) = 5, det(A₁) = 10, and det(A₂) = –5, then the solution is:
Analysis & Theory
x = det(A₁)/det(A) = 10/5 = 2, y = det(A₂)/det(A) = –5/5 = –1.
If a system of equations has infinitely many solutions, then in Cramer’s Rule:
B
det(A) = 0 and all det(Ai) = 0
C
det(A) = 0 and some det(Ai) ≠ 0
Analysis & Theory
For infinitely many solutions, det(A) = 0 and all det(Ai) = 0.
If a system of equations is inconsistent, then in Cramer’s Rule:
A
det(A) = 0 and some det(Ai) ≠ 0
C
det(A) = 0 and all det(Ai) = 0
Analysis & Theory
Inconsistency occurs if det(A) = 0 but some det(Ai) ≠ 0.
In solving a 3-variable system using Cramer’s Rule, A₂ is obtained by:
A
Replacing 1st column of A with constant column
B
Replacing 2nd column of A with constant column
C
Replacing 3rd column of A with constant column
D
Replacing diagonal of A with constant column
Analysis & Theory
A₂ is obtained by replacing the 2nd column of A with the constant column.
Cramer’s Rule expresses each variable as a:
A
Product of two determinants
B
Ratio of two determinants
C
Sum of two determinants
D
Difference of two determinants
Analysis & Theory
Each variable is given by det(Ai)/det(A), which is a ratio of determinants.
The advantage of Cramer’s Rule is:
A
It works for all systems regardless of det(A)
B
It gives exact solution using determinants
C
It is the fastest method for large systems
D
It avoids calculation of determinants
Analysis & Theory
Cramer’s Rule provides exact solution using determinant ratios, but is inefficient for large systems.