How to Find Rank of a 3×3 Matrix

The three methods to find rank of a 3×3 Matrix are:

Rank by Finding Minors
Rank of matrix by Echelon Form
Rank of matrix by Normal Form

Let’s discuss each method in detail.

Rank of Matrix by Finding Minors

Steps to find rank of 3×3 matrix by finding minors are given below.

Find the determinant of the matrix.
If Det(matrix) ≠ 0, then rank(matrix) = 3
If Det(matrix) = 0, then rank of matrix is the maximum order of one of the minors with non-zero value.

Example: Find the rank of 3×3 Matrix A = [Tex]\begin{bmatrix} -sin A&-cos A&1\\ cos A&sin A&1\\ 1&1& 0 \end{bmatrix}[/Tex] using minor method.

Solution:

First find the determinant of A

|A| = [Tex]\begin{vmatrix} -sin A&-cos A&1\\ cos A&sin A&1\\ 1&1& 0 \end{vmatrix}[/Tex]

|A| = (-sin A) × [0 -1] + cos A × [0- 1] + 1 × [cos A – sin A]

|A| = sin A – cos A + cos A – sin A

|A| = 0

Since, |A| = 0 we will determine maximum non-zero minor

Let’s take minor

M = [Tex]\begin{vmatrix} -sin A&1\\ 1& 0 \end{vmatrix}[/Tex]

M = (-sin A) × 0 – 1 × 1

M = -1

The maximum non-zero minor is of 2 × 2 .

So, Rank of matrix = 2

Rank of Matrix by Echelon Form

Steps to find rank of 3×3 matrix by finding Echelon form are given below.

Convert the matrix into its Echelon form using elementary row or column operations.
Then, find the number of non-zero rows in its Echelon form.
Rank(matrix) = Number of non-zero rows in Echelon form.

Example: Calculate the rank of matrix B = [Tex]\begin{bmatrix} 1&5&3\\ 2&7&3\\ -3&-6& 1 \end{bmatrix}[/Tex] using Echelon form.

Solution:

B = [Tex]\begin{bmatrix} 1&5&3\\ 2&7&3\\ -3&-6& 1 \end{bmatrix}[/Tex]

R₂← R₂ – 2R₁, R₃← R₃ + 3R₁

B = [Tex]\begin{bmatrix} 1&5&3\\ 0&-3&-3\\ 0&9& 10 \end{bmatrix}[/Tex]

R₂ ← R₂ / (-3)

B = [Tex]\begin{bmatrix} 1&5&3\\ 0&1&1\\ 0&9& 10 \end{bmatrix}[/Tex]

R₃ ← R₃ – 9R₂

B = [Tex]\begin{bmatrix} 1&5&3\\ 0&1&1\\ 0&0& 1 \end{bmatrix}[/Tex]

In the above matrix number of non-zero rows = 3

So, the rank of matrix B = 3

Rank of Matrix by Normal Form

Steps to find rank of 3×3 matrix by finding normal form are given below.

Convert the matrix in its normal form using the elementary row and column operation.
If [Tex]\begin{bmatrix} I_r & 0\\ 0&0 \end{bmatrix}[/Tex] be the normal form of the matrix.
Rank of matrix = r i.e., order of identity matrix.

Example: Calculate the rank of matrix P = [Tex]\begin{bmatrix} -1&-4&3\\ 2&8&-6\\ 3&12& -9 \end{bmatrix}[/Tex] using normal form.

Solution:

P = [Tex]\begin{bmatrix} -1&-4&3\\ 2&8&-6\\ 3&12& -9 \end{bmatrix}[/Tex]

C₂← C₂/ 4, C₃←C₃/ (-3)

P = [Tex]\begin{bmatrix} -1&-1&-1\\ 2&2&2\\ 3&3& 3 \end{bmatrix}[/Tex]

R₂← R₂/ 2, R₃←R₃/ 3

P = [Tex]\begin{bmatrix} -1&-1&-1\\ 1&1&1\\ 1&1&1 \end{bmatrix}[/Tex]

R₂← R₂ + R₁, R₃← R₃ + R₁

P = [Tex]\begin{bmatrix} -1&-1&-1\\ 0&0&0\\ 0&0&0 \end{bmatrix}[/Tex]

C₂← C₂– C₁, C₃← C₃– C₁

P = [Tex]\begin{bmatrix} -1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}[/Tex]

R₁← R₁/ (-1)

P = [Tex]\begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}[/Tex]

So, the normal form of matrix P is:

P = [Tex]\begin{bmatrix} I_1&0\\ 0&0\\ \end{bmatrix}[/Tex]

Rank of matrix P = 1

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