Concept understanding — Symmetric And Skew Symmetric Matrices
Symmetric and Skew-Symmetric Matrices
These are two special kinds of square matrices, defined by how a matrix compares with its own transpose A′ (the matrix with rows and columns swapped). They are among the most-tested ideas in the Matrices chapter.
Symmetric matrix
A square matrix A is symmetric if it equals its transpose:
A′=A,that isaij=aji for all i,j.
Entries are mirror images across the main diagonal. For example,
A=147425753,a12=a21=4,a13=a31=7.
Skew-symmetric matrix
A square matrix A is skew-symmetric if its transpose is its negative:
A′=−A,that isaij=−aji for all i,j.
Putting i=j gives aii=−aii, so 2aii=0 — every diagonal entry of a skew-symmetric matrix is 0. For example,
B=0−3230−5−250,bij=−bji.
Note
Both definitions demand a square matrix — the condition aij=±aji only makes sense when both entries exist.
Key facts
For any square matrix A, the matrix A+A′ is always symmetric and A−A′ is always skew-symmetric. (Check: (A+A′)′=A′+A=A+A′.)
If A is skew-symmetric of odd order, then detA=0. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
GUJCET 2026Set x1 markMCQ
Q.If A and B are skew-symmetric matrices of same order, then AB−BA is a ______
(A) Skew symmetric matrix
(B) Zero matrix
(C) Symmetric matrix
(D) Identity matrix
›Reveal solutionSolution
Take the transpose of AB−BA using AT=−A, BT=−B.
(AB−BA)T=(AB)T−(BA)T=BTAT−ATBT.
Since A and B are skew-symmetric, AT=−A and BT=−B:
Q.If a matrix A is both symmetric and skew-symmetric, then
(a) A is a diagonal matrix.
(b) A is a null (zero) matrix.
(c) A is a square matrix.
(d) None of these.
›Reveal solutionSolution
A matrix that is both symmetric and skew-symmetric must equal its own negative, forcing every entry to be zero.
If A is symmetric, AT=A. If A is skew-symmetric, AT=−A. Combining, A=−A, so 2A=O, giving A=O (the null matrix). Note this also forces A to be square, but option (c) alone is not the s …