Exercise · ZW-MATRIX-RANK-009 · ★★★ Advanced
Compute the rank of \(\begin{pmatrix}1&2&1\\2&4&3\\1&2&2\end{pmatrix}\).
Choose one option.
Show answer
- B. 2
Explanation
With \(R_2-2R_1=(0,0,1)\) and \(R_3-R_1=(0,0,1)\), the last row vanishes after subtracting. There are 2 pivots (columns 1 and 3): rank 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?