Question:
If
\[
A = \begin{bmatrix}0 & 1 & 2 & 3 \\ 0 & 2 & 4 & 8 \\ 0 & 0 & 4 & 8\end{bmatrix},
\text{ then the rank of } A \text{ is:}
\]
If
\[
A = \begin{bmatrix}0 & 1 & 2 & 3 \\ 0 & 2 & 4 & 8 \\ 0 & 0 & 4 & 8\end{bmatrix},
\text{ then the rank of } A \text{ is:}
\]
Show Hint
The rank of a matrix is equal to the maximum number of linearly independent rows or columns.
Updated On: Jul 18, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Check the rows of \(A\).
\[ R_1 = [0,1,2,3],\quad R_2 = [0,2,4,8],\quad R_3 = [0,0,4,8] \]
Step 2: Row reductions.
\(-2R_1 + R_2 \Rightarrow R_2 \to [0,0,0,2]\)
R3 = [0,0,4,8]
Step 3: Check linear independence.
- \(R_1\) and new \(R_2\) are independent.
- \(R_3\) is a combination of \(R_1\) and \(R_2\), so not independent.
Step 4: Conclusion.
Number of linearly independent rows = 2
\[ \boxed{\text{rank}(A) = 2} \]
\[ R_1 = [0,1,2,3],\quad R_2 = [0,2,4,8],\quad R_3 = [0,0,4,8] \]
Step 2: Row reductions.
\(-2R_1 + R_2 \Rightarrow R_2 \to [0,0,0,2]\)
R3 = [0,0,4,8]
Step 3: Check linear independence.
- \(R_1\) and new \(R_2\) are independent.
- \(R_3\) is a combination of \(R_1\) and \(R_2\), so not independent.
Step 4: Conclusion.
Number of linearly independent rows = 2
\[ \boxed{\text{rank}(A) = 2} \]
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