Exercise · ZW-TRANSPOSE-MATRIX-003 · ★☆☆ Basic
Which of these matrices is symmetric (satisfies \(A=A^T\))?
Choose one option.
Show answer
- C. \(\begin{pmatrix}1 & 2 \\ 2 & 3\end{pmatrix}\)
Explanation
A matrix is symmetric if \(a_{ij}=a_{ji}\). In \(\begin{pmatrix}1 & 2 \\ 2 & 3\end{pmatrix}\), \(a_{12}=a_{21}=2\) and it equals its transpose. The one with \(1\) and \(-1\) off the diagonal flips sign (skew-symmetric) and the \(2\times3\) one is not even square.
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
- ZW-TRANSPOSE-MATRIX-001 Compute the transpose of [1 2 3; 4 5 6].
- ZW-TRANSPOSE-MATRIX-002 If A is a 2×4 matrix, what size is 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?
- ZW-TRANSPOSE-MATRIX-009 A square matrix satisfies A^T=-A (skew-symmetric). What can be said about its main diagonal?