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What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?

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Question: What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?

Options:

  1. 1
  2. 2
  3. 3
  4. 0

Correct Answer: 2

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.

What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?

Practice Questions

Q1
What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
  1. 1
  2. 2
  3. 3
  4. 0

Questions & Step-by-Step Solutions

What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
  • 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].
  • Step 8: Therefore, the rank of the matrix E is 2.
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