Exercise · ZW-TRANSPOSE-MATRIX-007 · ★★☆ Intermediate
Compute \(A+A^T\) for \(A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}\).
Choose one option.
Show answer
- B. \(\begin{pmatrix}2 & 5 \\ 5 & 8\end{pmatrix}\)
Explanation
\(A^T=\begin{pmatrix}1 & 3\\ 2 & 4\end{pmatrix}\), so the sum is \(\begin{pmatrix}2 & 5\\ 5 & 8\end{pmatrix}\). It is always symmetric: \((A+A^T)^T=A^T+A\).
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-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-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?