Examples on Rank and Nullity
Some examples on rank and nullity are,
Example 1: Given Matrix
[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]
Find the rank and nullity of B.
Solution:
[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]
Using Row Transformation in matrix B,
R2 → R3 – 2R2
[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]
Now, R3 → R3 – 4R1
[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & -3 & 3 & 9 \\ \end{pmatrix}[/Tex]
Now, R3 → 3R2 + R3
[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & 0 & 0 & 0 \\ \end{pmatrix}[/Tex]
∴ r (B) = 2.
n (B) = n (columns) – r (B) = 4 – 2 = 2.
∴ Rank of matrix B is 2 and the nullity of matrix B is 2.
Example 2: Given Matrix
[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\-1 & -5 & -1 & 8 \\ \end{pmatrix}[/Tex]
Find the rank of matrix A.
Solution:
[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\-1 & -5 & -1 & 8 \\ \end{pmatrix}[/Tex]
Using Row Transformation in matrix A,
R3 → R3 + R1
[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\0 & -7 & -1 & 12 \\ \end{pmatrix}[/Tex]
Now, R2 → R2 – 3R1
[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\0 & 7 & 1 & -12 \\0 & -7 & -1 & 12 \\ \end{pmatrix}[/Tex]
Now, R3 → R3 + R2
[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\0 & 7 & 1 & -12 \\0 & 0 & 0 & 0 \\ \end{pmatrix}[/Tex]
r (A) = 2
∴ Rank of matrix A is 2.
Example 3: Given Matrix
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}[/Tex]
Find the nullity of matrix D.
Solution:
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}[/Tex]
Using Row Transformation in matrix D,
R3 → R3 – 5R1
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & -16 \\-2 & 3 \\ \end{pmatrix}[/Tex]
Now, R4 → 2R1 + R4
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & -16 \\0 & 9 \\ \end{pmatrix}[/Tex]
Now, R3 → -8R2 + R3
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 9 \\ \end{pmatrix}[/Tex]
Now, R4 → 9R2 + 2R4
[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 0 \\ \end{pmatrix}[/Tex]
Now, R2 → -1/2 R2
[Tex]D = \begin{pmatrix} 1 & 3\\0 & 1 \\0 & 0 \\0 & 0 \\ \end{pmatrix}[/Tex]
r (D) = 2
n (D) = n (columns) – r (D) = 2 – 2 = 0.
∴ Nullity of matrix D is 0.
Rank and Nullity
Rank and Nullity are essential concepts in linear algebra, particularly in the context of matrices and linear transformations. They help describe the number of linearly independent vectors and the dimension of the kernel of a linear mapping.
In this article, we will learn what Rank and Nullity, the Rank-Nullity Theorem, and their applications, advantages, and limitations.
Table of Content
- What is Rank and Nullity?
- Calculating Rank and Nullity
- Rank-Nullity Theorem
- Rank-Nullity Theorem Proof
- Advantages of Rank and Nullity
- Application of Rank and Nullity
- Limitations of Rank and Nullity
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