A matrix A is skew-symmetric if
A
A = A^T
B
A = -A^T
C
A^T = I
D
A^2 = A
Analysis & Theory
By definition, a skew-symmetric matrix satisfies A^T = -A.
A skew-symmetric matrix must always be
A
Square
B
Rectangular
C
Diagonal
D
Upper triangular
Analysis & Theory
Only square matrices can be skew-symmetric since A and A^T must be of the same order.
The diagonal elements of a skew-symmetric matrix are always
A
Zero
B
One
C
Equal to each other
D
Non-zero
Analysis & Theory
For a_ii = -a_ii ⇒ 2a_ii = 0, hence diagonal elements must be zero.
If A = [[0, 2], [-2, 0]], then A is
A
Symmetric
B
Skew-symmetric
C
Diagonal
D
Identity
Analysis & Theory
Since A^T = [[0, -2], [2, 0]] = -A, the matrix is skew-symmetric.
If A = [[0, -3], [3, 0]], then A is
A
Symmetric
B
Skew-symmetric
C
Diagonal
D
Triangular
Analysis & Theory
Transpose gives [[0, 3], [-3, 0]] which equals -A.
In a skew-symmetric matrix, the relation between elements is
A
a_ij = a_ji
B
a_ij = -a_ji
C
a_ij = 0
D
a_ij = 1
Analysis & Theory
For skew-symmetric: a_ij = -a_ji for all i, j.
The sum of two skew-symmetric matrices of the same order is
A
Always symmetric
B
Always skew-symmetric
C
Always diagonal
D
Zero matrix
Analysis & Theory
If A^T = -A and B^T = -B, then (A+B)^T = A^T + B^T = -(A+B).
The difference of two skew-symmetric matrices of the same order is
A
Always symmetric
B
Always skew-symmetric
C
Always triangular
D
Zero matrix
Analysis & Theory
If A and B are skew-symmetric, then (A−B)^T = A^T − B^T = −A − (−B) = −(A−B).
Which of the following is always skew-symmetric?
A
A + A^T
B
A − A^T
C
AA^T
D
A^2
Analysis & Theory
A − A^T is always skew-symmetric because (A − A^T)^T = −(A − A^T).
The determinant of every skew-symmetric matrix of odd order is
A
Zero
B
One
C
Non-zero
D
Equal to its trace
Analysis & Theory
For odd-order skew-symmetric matrices, determinant is always zero.