If I = [[1, 1], [1, 1]], what is the rank of I? (2022)

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Question: If I = [[1, 1], [1, 1]], what is the rank of I? (2022)

Options:

Correct Answer: 1

Exam Year: 2022

Solution:

The rank of I is 1 because all rows are linearly dependent.

If I = [[1, 1], [1, 1]], what is the rank of I? (2022)

If I = [[1, 1], [1, 1]], what is the rank of I? (2022)

Step 1: Understand what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns.
Step 2: Identify the given matrix I, which is [[1, 1], [1, 1]]. This matrix has 2 rows and 2 columns.
Step 3: Define what rank means. The rank of a matrix is the maximum number of linearly independent rows or columns.
Step 4: Look at the rows of matrix I. The first row is [1, 1] and the second row is also [1, 1].
Step 5: Check if the rows are linearly independent. Two rows are linearly dependent if one row can be expressed as a multiple of the other.
Step 6: Since the second row [1, 1] is exactly the same as the first row [1, 1], they are linearly dependent.
Step 7: Since both rows are dependent, we can only count one of them as independent.
Step 8: Therefore, the rank of matrix I is 1.

Matrix Rank – The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix.
Linear Dependence – Rows or columns are linearly dependent if one can be expressed as a linear combination of others.