The matrix inversion method is applicable when:
A
Coefficient matrix is singular
B
Determinant of coefficient matrix is non-zero
C
System is inconsistent
D
All of the above
Analysis & Theory
Matrix inversion method works only if det(A) ≠ 0, i.e., A is invertible (non-singular).
In the matrix inversion method, the system of equations AX = B is solved as:
A
X = AB
B
X = A⁻¹B
C
X = BA⁻¹
D
X = det(A)/B
Analysis & Theory
We multiply both sides by A⁻¹ to get X = A⁻¹B.
If det(A) = 0, then:
A
Matrix inversion method can still be applied
B
A⁻¹ does not exist
C
System always has unique solution
D
System is always homogeneous
Analysis & Theory
If det(A) = 0, A⁻¹ does not exist, hence inversion method cannot be applied.
The adjoint method is used to find:
A
Determinant of a matrix
B
Inverse of a matrix
C
Rank of a matrix
D
Trace of a matrix
Analysis & Theory
Inverse of a matrix is calculated using A⁻¹ = adj(A)/det(A).
In solving a 2×2 system using inversion method, if A = [[2,1],[1,3]] and B = [[5],[7]], then X = ?
A
[1,2]
B
[2,1]
C
[3,1]
D
[1,3]
Analysis & Theory
det(A)=5, adj(A)=[[3,-1],[-1,2]], A⁻¹ = (1/5)[[3,-1],[-1,2]], X = A⁻¹B = [2,1].
Which of the following is a disadvantage of the matrix inversion method?
A
It cannot be applied when det(A) ≠ 0
B
It is computationally heavy for large systems
C
It always gives approximate results
D
It requires eigenvalues
Analysis & Theory
Matrix inversion method is exact but becomes inefficient for large systems due to heavy calculations.
For the system AX = B, if A⁻¹ exists, then the solution is:
A
Unique
B
Infinite
C
No solution
D
Trivial only
Analysis & Theory
If A⁻¹ exists, the system has a unique solution X = A⁻¹B.
The inverse of a square matrix A exists only if:
A
Rank(A) < n
B
det(A) = 0
C
det(A) ≠ 0
D
A is triangular
Analysis & Theory
A square matrix has an inverse if and only if its determinant is non-zero.
Which relation is true for an invertible matrix A?
A
AA⁻¹ = I
B
A⁻¹A = I
C
Both AA⁻¹ = I and A⁻¹A = I
D
A² = I
Analysis & Theory
By definition, both AA⁻¹ = I and A⁻¹A = I hold for an invertible matrix.
The matrix inversion method is best suited for:
A
Small systems of equations
B
Very large systems
C
Systems with infinite solutions
D
Inconsistent systems
Analysis & Theory
Matrix inversion method is efficient for small systems but impractical for large ones.