What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
Practice Questions
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Q1
What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
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The rank of a matrix is the maximum number of linearly independent row vectors. Here, the first and third rows are independent, while the second row is zero. Thus, the rank is 2.
Questions & Step-by-step Solutions
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Q
Q: What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
Solution: The rank of a matrix is the maximum number of linearly independent row vectors. Here, the first and third rows are independent, while the second row is zero. Thus, the rank is 2.
Steps: 8
Step 1: Identify the matrix E, which is E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]].
Step 2: Look at the rows of the matrix. The rows are: [1, 2, 3], [0, 0, 0], and [4, 5, 6].
Step 3: Determine if the rows are linearly independent. A row is linearly independent if it cannot be formed by a combination of other rows.
Step 4: The first row [1, 2, 3] is independent because it is not a zero row.
Step 5: The second row [0, 0, 0] is a zero row and does not contribute to the rank.
Step 6: The third row [4, 5, 6] is also independent because it cannot be formed by the first row.
Step 7: Count the number of independent rows. We have two independent rows: [1, 2, 3] and [4, 5, 6].
