A matrix A is symmetric if
A
A = A^T
B
A = -A^T
C
A^T = I
D
A^2 = A
Analysis & Theory
By definition, a symmetric matrix is equal to its transpose.
A symmetric matrix must always be
A
Square
B
Rectangular
C
Diagonal
D
Zero matrix
Analysis & Theory
Only square matrices can be symmetric since A and A^T must have the same order.
If A = [[2, 3], [3, 5]], then A is
A
Symmetric
B
Skew-symmetric
C
Diagonal
D
Identity
Analysis & Theory
Since A^T = [[2, 3], [3, 5]] = A, it is symmetric.
If A = [[0, -1], [-1, 0]], then A is
A
Symmetric
B
Skew-symmetric
C
Identity
D
Diagonal
Analysis & Theory
Since A^T = A, this is symmetric, even though it has negative elements.
In a symmetric matrix, elements satisfy the condition
A
a_ij = -a_ji
B
a_ij = a_ji
C
a_ij = 0
D
a_ij = 1
Analysis & Theory
For symmetry: a_ij = a_ji for all i and j.
Which of the following is symmetric?
A
[[1, 2], [3, 4]]
B
[[2, -1], [-1, 3]]
C
[[0, 1], [-1, 0]]
D
[[5, 0], [0, 5]]
Analysis & Theory
[[2, -1], [-1, 3]] is symmetric because its transpose is the same.
The sum of two symmetric matrices of the same order is
A
Always symmetric
B
Always skew-symmetric
C
Always diagonal
D
None of these
Analysis & Theory
If A = A^T and B = B^T, then (A+B)^T = A^T + B^T = A + B.
The difference of two symmetric matrices of the same order is
A
Always symmetric
B
Always skew-symmetric
C
Always diagonal
D
Always zero
Analysis & Theory
If A = A^T and B = B^T, then (A−B)^T = A^T − B^T = A − B.
The product of two symmetric matrices A and B is symmetric if and only if
A
AB = BA
B
AB = -BA
C
A = B
D
A = -B
Analysis & Theory
For product AB to be symmetric, AB must equal BA.
Which of the following is always symmetric?
A
A - A^T
B
A + A^T
C
AA^T - A^T A
D
None
Analysis & Theory
A + A^T is always symmetric because (A + A^T)^T = A + A^T.