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Matrix Rank Calculation Guide

The rank of a matrix is defined as the order of the largest non-zero minor (determinant of a submatrix). To find the rank of a matrix: 1) Find the largest square submatrix and calculate its determinant 2) The rank is equal to the size of the largest submatrix whose determinant is non-zero 3) An alternative method is to row reduce the matrix into echelon form, where the number of non-zero rows is equal to the rank.

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102 views4 pages

Matrix Rank Calculation Guide

The rank of a matrix is defined as the order of the largest non-zero minor (determinant of a submatrix). To find the rank of a matrix: 1) Find the largest square submatrix and calculate its determinant 2) The rank is equal to the size of the largest submatrix whose determinant is non-zero 3) An alternative method is to row reduce the matrix into echelon form, where the number of non-zero rows is equal to the rank.

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dynamiccareedu
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Rank of a Matrix

Minor of Matrix

If any 𝑟 rows and any 𝑟 columns from an 𝑚 × 𝑛 matrix 𝐴 are retained and remaining (𝑚 −
𝑟) rows and (𝑛 − 𝑟) columns removed, then the determinant of the remaining 𝑟 × 𝑟
submatrix of 𝐴 is called minor of 𝐴 of order 𝑟.

Rank of matrix

If in an 𝑚 × 𝑛 matrix 𝐴, at least one of its 𝑟 × 𝑟 minors is different from zero while all the
minors of order (𝑟 + 1) are zero, then 𝑟 is defined as the rank of the matrix 𝐴.

In other words, Rank of a matrix is defined as the order of the largest square sub matrix, of
the given matrix, whose determinant is not zero. Denoted as ‘𝜌(𝐴)’.

Finding of Rank of a matrix

Using sub-matrix/mirors Forming Echelon form

2 3 1
e.g. (
A= 4 5 6)
2 3 9
2 3
A₁ = ( )
4 5
A₁ ⸦ A

Finding Rank

Step 1: Find largest sub matrix.

Step 2: Find the determinant not zero.

2 3 1
|𝐴| = (4 5 6)
2 3 9
= 2(45 − 18)

= 2 ∗ 27

= 54
|𝐴| = 54

≠0

𝜌(𝐴) = 3

1 2 3
PROB: 𝐵 = (2 4 7 ) 𝜌(𝐵) = ?
3 6 10

SOLUTION:

The largest order of square sub matrix is 3 × 3

Now, |𝐵| = 1(40 − 42) − 2(20 − 21) + 3(12 − 12)

= −2 + 2

=0

|𝐵₁| = |1 2|
2 4
=4−4

=0

|𝐵₂| = |4 7 |
6 10
= 40 − 42

= −2

≠0

4 7
As 𝐵 2 = ( ) which is order of 2 × 2 is of determinant not zero hence the rank of the
6 10
given matrix is 𝜌(𝐵) = 2.

Echelon Matrices
A matrix A is called an echelon matrix, or is said to be in echelon form, if the following two
conditions hold (where a leading nonzero element of a row of A is the first nonzero element
in the row):
(1) All zero rows, if any, are at the bottom of the matrix.
(2) Each leading nonzero entry in a row is to the right of the leading nonzero entry in the
preceding row.

Forming Echelon form

1 2 3
PROB: 𝐴 = (2 4 7 ) to find rank of A by forming Echelon form.
3 6 10

1 2 3
SOLUTION: 𝐴 = (2 4 7)
3 6 10
1 2 3
≈ (0 0 −1) 𝑅2′ = 2 𝑅₁ − 𝑅₂
0 0 −1
𝑅′₃ = 3𝑅₁ − 𝑅₃

1 2 3
≈( )
0 0 −1

In this Echelon form, there are two non-zero rows, hence the rank of the given matrix is
𝜌(𝐴) = 2.

1 2 3
* 𝐴 = (2 5 8)
4 10 18

1 2 3
≈ (0 0 −2)
0 0 −2

1 3 4 3
* 𝐶 = (3 9 12 4)
1 3 4 1

1 3 4 3
≈ (0 0 0 5)
0 0 0 2
1 3 4 3
≈ (0 0 0 5)
0 0 0 0

In this Echelon form there is 2 pivot elements so, the rank of the matrix 𝜌(𝐴) = 2.

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