1. What is Rank and Nullity?

The rank of a matrix is the dimension of the vector space generated by its columns (column rank) or rows (row rank). The nullity is the dimension of the kernel (null space) of the matrix. According to the Rank-Nullity Theorem, for any matrix A with n columns: rank(A) + nullity(A) = n.

2. How Does the Calculator Work?

The calculator performs the following steps:

1. Converts the input matrix to Reduced Row Echelon Form (RREF)
2. Counts the number of non-zero rows in RREF to determine rank
3. Calculates nullity using the Rank-Nullity Theorem

3. Importance of Rank and Nullity

Details: Rank determines the number of linearly independent rows/columns and thus the dimension of the image space. Nullity indicates the dimension of the solution space for the homogeneous system Ax=0. These concepts are fundamental in linear algebra with applications in solving systems of equations, linear transformations, and more.

4. Using the Calculator

Tips: Enter your matrix with elements separated by spaces or commas, and rows separated by newlines or semicolons. For example:
1 2 3; 4 5 6; 7 8 9 or
1, 2, 3 4, 5, 6 7, 8, 9

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rank and nullity?
A: Rank is the dimension of the column space (number of linearly independent columns), while nullity is the dimension of the null space (number of free variables in the solution to Ax=0).

Q2: Can rank exceed the number of rows or columns?
A: No, rank cannot exceed either the number of rows or columns of the matrix.

Q3: What does nullity=0 mean?
A: Nullity=0 means the matrix has a trivial null space (only the zero vector), indicating the columns are linearly independent.

Q4: How is rank related to invertibility?
A: A square matrix is invertible if and only if its rank equals its dimension (full rank).

Q5: Does row operations change rank?
A: Elementary row operations don't change the rank of a matrix, which is why we can use RREF to determine rank.