A square matrix A is said to have a multiplicative inverse if
A
det(A) = 0
B
det(A) ≠ 0
C
A is symmetric
D
A is diagonal
Analysis & Theory
A matrix is invertible (non-singular) only if det(A) ≠ 0.
The inverse of a square matrix A is denoted as
A
Aᵀ
B
adj(A)
C
A⁻¹
D
|A|
Analysis & Theory
By convention, the inverse of A is written as A⁻¹.
The condition for existence of inverse of a square matrix A is
A
det(A) = 0
B
det(A) ≠ 0
C
A is diagonal
D
A is symmetric
Analysis & Theory
Inverse exists only for non-singular matrices, i.e., det(A) ≠ 0.
If A⁻¹ exists, then the relation between A and A⁻¹ is
A
AA⁻¹ = I
B
AA⁻¹ = 0
C
AA⁻¹ = A
D
AA⁻¹ = det(A)
Analysis & Theory
By definition, AA⁻¹ = A⁻¹A = I (identity matrix).
The formula to find the inverse of a matrix A is
A
A⁻¹ = det(A)/adj(A)
B
A⁻¹ = adj(A)/det(A)
C
A⁻¹ = transpose(A)/det(A)
D
A⁻¹ = cofactor(A)/det(A)
Analysis & Theory
Inverse formula: A⁻¹ = adj(A)/det(A), provided det(A) ≠ 0.
If A is a 2×2 matrix [[a, b], [c, d]], then A⁻¹ is
A
(1/(ad − bc)) × [[d, -b], [-c, a]]
B
(1/(ad − bc)) × [[a, b], [c, d]]
C
(ad − bc) × [[d, -b], [-c, a]]
D
[[d, -b], [-c, a]] only
Analysis & Theory
For 2×2: A⁻¹ = (1/det(A)) × adj(A).
If A is an invertible matrix, then (A⁻¹)⁻¹ equals
A
A
B
I
C
0
D
adj(A)
Analysis & Theory
Inverse of the inverse gives back the original matrix: (A⁻¹)⁻¹ = A.
If A and B are invertible matrices of same order, then (AB)⁻¹ =
A
A⁻¹B⁻¹
B
B⁻¹A⁻¹
C
AB
D
None of the above
Analysis & Theory
(AB)⁻¹ = B⁻¹A⁻¹ (order is reversed).
If det(A) = 5, then det(A⁻¹) equals
A
5
B
1/5
C
-5
D
0
Analysis & Theory
Property: det(A⁻¹) = 1/det(A).
If A is orthogonal, then A⁻¹ equals
A
adj(A)
B
det(A)
C
Aᵀ
D
−A
Analysis & Theory
For orthogonal matrices, A⁻¹ = Aᵀ.