This is a fundamental question in linear algebra about the properties of the rank of a non-square matrix.
Step 1: Understanding the Question:
We need to determine all possible integer values that the rank of a matrix with 4 rows and 3 columns can take.
Step 2: Key Formula or Approach:
The rank of a matrix is defined as the maximum number of linearly independent columns (or, equivalently, rows). A key property is that the rank of an \(m \times n\) matrix \(A\) is bounded by both \(m\) and \(n\).
\[ \text{rank}(A) \le \min(m, n) \]
Step 3: Detailed Explanation:
Identify m and n: For the given \(4 \times 3\) matrix, we have \(m=4\) rows and \(n=3\) columns.
Apply the rank inequality: The rank must be less than or equal to the minimum of the number of rows and columns.
\[ \text{rank}(A) \le \min(4, 3) = 3 \]
This means the rank cannot be 4.
Consider the minimum rank: The smallest possible rank for any matrix is 0. This occurs if and only if the matrix is the zero matrix (all entries are zero).
Combine the bounds: The rank must be an integer that is greater than or equal to 0 and less than or equal to 3.
Thus, the possible values for the rank are 0, 1, 2, and 3. Each of these values is achievable with an appropriately constructed \(4 \times 3\) matrix.
Step 4: Final Answer:
The possible values for the rank are 0, 1, 2, 3.