Exercise · ZW-TRANSPOSE-MATRIX-010 · ★★★ Advanced
Let \(A\) be an \(m\times n\) matrix. What can be said about \(A^TA\)?
Choose one option.
Show answer
- B. It is \(n\times n\) and symmetric.
Explanation
\(A^T\) is \(n\times m\) and \(A\) is \(m\times n\), so \(A^TA\) is \(n\times n\). Also \((A^TA)^T=A^T(A^T)^T=A^TA\), hence it is symmetric.
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-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?