- #1
underacheiver
- 12
- 0
Homework Statement
Find the rank of A =
{[1 0 2 0]
[4 0 3 0]
[5 0 -1 0]
[2 -3 1 1]}
Homework Equations
The Attempt at a Solution
i row reduced A to be:
{[1 0 0 0]
[0 1 0 -1/3]
[0 0 1 0]}
where do i go from here?
underacheiver said:The column rank of a matrix A is the maximal number of linearly independent columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A.
Since the column rank and the row rank are always equal, they are simply called the rank of A.
underacheiver said:so is it 3? because the 2nd and 4th columns are dependent.
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It is a measure of the dimension or "size" of the vector space spanned by the rows or columns of the matrix.
The rank of a matrix can be determined by performing row reduction operations on the matrix and counting the number of non-zero rows or columns in the reduced matrix. Alternatively, it can also be determined by finding the number of pivot columns in the matrix after performing Gaussian elimination.
The rank of a matrix is important because it tells us how many linearly independent rows or columns exist in the matrix. It also determines the dimension of the vector space spanned by the rows or columns of the matrix, which can have implications in applications such as data analysis and machine learning.
Yes, the rank of a matrix can change if the matrix undergoes a linear transformation. However, the rank of a matrix is an inherent property and does not change based on row or column operations.
A square matrix is invertible if and only if its rank is equal to its dimension. This means that a matrix with full rank (rank equal to its dimension) will have an inverse, while a matrix with less than full rank will not have an inverse.