What is the rank of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2023)

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Question: What is the rank of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2023)

Options:

Correct Answer: 2

Exam Year: 2023

Solution:

The rank of E is 2 because the rows are linearly dependent (the third row is a linear combination of the first two).

What is the rank of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2023)

What is the rank of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2023)

Step 1: Write down the matrix E: [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Step 2: Identify the number of rows and columns in the matrix. E has 3 rows and 3 columns.
Step 3: Check if the rows are linearly independent. This means we need to see if one row can be formed by combining the others.
Step 4: Look at the third row [7, 8, 9]. Notice that it can be formed by adding the first row [1, 2, 3] multiplied by 3 and the second row [4, 5, 6] multiplied by 0. This shows that the third row is dependent on the first two rows.
Step 5: Since the third row is dependent, we only have 2 independent rows: [1, 2, 3] and [4, 5, 6].
Step 6: The rank of a matrix is the number of linearly independent rows. Here, we have 2 independent rows.
Step 7: Therefore, the rank of the matrix E is 2.

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