Exercise · ZW-TRANSPOSE-MATRIX-009 · ★★★ Advanced
A square matrix satisfies \(A^T=-A\) (skew-symmetric). What can be said about its main diagonal?
Choose one option.
Show answer
- A. All diagonal entries are \(0\).
Explanation
On the diagonal, \(a_{ii}=-a_{ii}\) (the entry does not move under transposition), which forces \(2a_{ii}=0\), i.e. \(a_{ii}=0\). The other entries satisfy \(a_{ji}=-a_{ij}\) and may be nonzero.
Review the concept: Transpose matrix
The transpose \(A^T\) of a matrix \(A\) is the one obtained by exchanging its rows and columns.
\[(A^T)_{ij}=a_{ji}\]
More practice problems: transpose matrix
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- ZW-TRANSPOSE-MATRIX-003 Which of these matrices is symmetric (satisfies A=A^T)?
- ZW-TRANSPOSE-MATRIX-004 If A^T=[1 3; 2 4], what is A?
- ZW-TRANSPOSE-MATRIX-005 If A and B have the same size, what is (A+B)^T equal to?
- ZW-TRANSPOSE-MATRIX-006 With A=[1 2; 0 1] and B=[3 0; 1 2], compute (AB)^T.
- ZW-TRANSPOSE-MATRIX-007 Compute A+A^T for A=[1 2; 3 4].
- ZW-TRANSPOSE-MATRIX-008 Let x=(1,-2,3)^T be a column vector. What is x^Tx?