The adjoint of a square matrix A is defined as
A
Transpose of A
B
Inverse of A
C
Transpose of cofactor matrix of A
D
Determinant of A
Analysis & Theory
Adjoint of A is the transpose of its cofactor matrix.
If A is a square matrix, then A × adj(A) =
A
O (zero matrix)
B
I (identity matrix)
C
det(A) × I
D
Cofactor matrix
Analysis & Theory
Property: A·adj(A) = adj(A)·A = det(A)·I.
If A is a non-singular matrix, then A⁻¹ can be expressed as
A
adj(A)/det(A)
B
det(A)/adj(A)
C
transpose(A)/det(A)
D
cofactor(A)/det(A)
Analysis & Theory
Inverse: A⁻¹ = adj(A)/det(A) if det(A) ≠ 0.
If A is singular, then adj(A) is
A
Zero matrix (always)
B
May or may not be zero
C
Identity matrix
D
Transpose of A
Analysis & Theory
If det(A)=0, A is singular, and adj(A) may be zero or non-zero.
The order of adjoint of an n×n matrix is
A
n×n
B
(n−1)×(n−1)
C
n×(n−1)
D
Depends on determinant
Analysis & Theory
Adjoint of an n×n matrix is also n×n.
If A is a 2×2 matrix [[a, b], [c, d]], then adj(A) is
A
[[d, -b], [-c, a]]
B
[[a, b], [c, d]]
C
[[a, -b], [-c, d]]
D
[[c, d], [a, b]]
Analysis & Theory
For 2×2: adj(A) = [[d, -b], [-c, a]].
For an identity matrix I, adj(I) equals
A
Zero matrix
B
Identity matrix
C
Diagonal matrix with det(I)
D
Transpose of I
Analysis & Theory
Adjoint of identity matrix is itself.
If A is a diagonal matrix, then adj(A) is
A
Transpose of A
B
Diagonal with product of elements divided by each diagonal element
C
Zero matrix
D
Identity matrix
Analysis & Theory
Adjoint of diagonal matrix has diagonal entries = product of all diagonal elements / that element.
If det(A) = 0, then A·adj(A) equals
A
Zero matrix
B
Identity matrix
C
Adj(A)
D
Transpose of A
Analysis & Theory
Since det(A)=0 ⇒ A·adj(A) = 0 matrix.
The relation between adj(Aᵀ) and adj(A) is
A
adj(Aᵀ) = adj(A)
B
adj(Aᵀ) = (adj(A))ᵀ
C
adj(Aᵀ) = det(A)
D
adj(Aᵀ) = A
Analysis & Theory
Property: adj(Aᵀ) = (adj(A))ᵀ.