Rank and Nullity Calculator

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Example

Created on 2024-06-20Asked by Jackson White (Solvelet student)

Find the rank and nullity of the matrix:

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}.

Solution

To find the rank and nullity of the matrix

A

: 1. **Row Reduce the Matrix:** Perform row operations to get the matrix into row echelon form:

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}.

R_2 \rightarrow R_2 - 4R_1 \quad \Rightarrow \quad \begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 7 & 8 & 9 \end{pmatrix}.

R_3 \rightarrow R_3 - 7R_1 \quad \Rightarrow \quad \begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \end{pmatrix}.

R_3 \rightarrow R_3 - 2R_2 \quad \Rightarrow \quad \begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 0 \end{pmatrix}.

2. **Determine the Rank:** The rank of a matrix is the number of non-zero rows in its row echelon form:

\text{Rank}(A) = 2.

3. **Calculate the Nullity:** The nullity of a matrix is the number of columns minus the rank:

\text{Nullity}(A) = \text{Number of columns} - \text{Rank}(A) = 3 - 2 = 1.

Therefore, the rank of the matrix is 2, and the nullity is 1. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Levi Taylor on Solvelet

A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 2 & 4 & 6 \end{bmatrix}

B = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 2 & 3 \\ 0 & 1 & 4 \end{bmatrix}

Definition

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