Exercise · ZW-TRANSPOSE-MATRIX-008 · ★★☆ Intermediate
Let \(\mathbf{x}=(1,-2,3)^T\) be a column vector. What is \(\mathbf{x}^T\mathbf{x}\)?
Choose one option.
Show answer
- A. \(14\)
Explanation
\(\mathbf{x}^T\) is a \(1\times3\) row vector, so \(\mathbf{x}^T\mathbf{x}\) is \(1\times1\): \(1+4+9=14\). The \(3\times3\) matrix shown as a distractor is \(\mathbf{x}\mathbf{x}^T\).
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-009 A square matrix satisfies A^T=-A (skew-symmetric). What can be said about its main diagonal?