Exercise · ZW-MATRIX-RANK-010 · ★★★ Advanced
For which value of \(k\) does \(\begin{pmatrix}1&2\\3&k\end{pmatrix}\) have rank 1?
Choose one option.
Show answer
- C. 6
Explanation
The rank is 1 when the second row is a multiple of the first: \((3,k)=3(1,2)\) gives \(k=6\). For any other \(k\) the rows are independent and the rank is 2.
Review the concept: Rank of a matrix
The rank of a matrix \(A\) is the dimension of the subspace spanned by its columns.
\[\operatorname{rank}(A)=\dim\operatorname{col}(A)\]
More practice problems: rank of a matrix
- ZW-MATRIX-RANK-001 What is the rank of [1 2; 2 4]?
- ZW-MATRIX-RANK-002 What is the rank of the identity matrix I_3?
- ZW-MATRIX-RANK-003 What is the largest possible rank of a 3× 5 matrix?
- ZW-MATRIX-RANK-004 What is the rank of the 2× 2 zero matrix?
- ZW-MATRIX-RANK-005 Compute the rank of [1 2 3; 4 5 6; 7 8 9].
- ZW-MATRIX-RANK-006 Compute the rank of [1 0 2; 0 1 3; 1 1 5].
- ZW-MATRIX-RANK-007 At most how many linearly independent rows does A=[1 2 0 1; 0 0 1 3; 1 2 1 4] have?
- ZW-MATRIX-RANK-008 A 4× 4 matrix A has rank 3. Which statement is true?