Exercise · ZW-MATRIX-RANK-008 · ★★☆ Intermediate
A \(4\times 4\) matrix \(A\) has rank 3. Which statement is true?
Choose one option.
Show answer
- B. The columns of \(A\) span a 3-dimensional subspace of \(\mathbb{R}^4\).
Explanation
The rank is the dimension of the span of the columns: 3 inside \(\mathbb{R}^4\). With rank below 4 the columns are dependent, \(A\) is not invertible and some \(b\) lie outside the column space.
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-009 Compute the rank of [1 2 1; 2 4 3; 1 2 2].