Find the rank of the matrix \(\left(\begin{matrix} 8 & 1 & 3 & 6 \\\ 0 & 3 & 2 & 2 \\\ -8 & -1 & -3 & -4 \end{matrix}\right)\)

Concept:

Rank:

The rank of a matrix is a number equal to the order of the highest order non-vanishing minor, that can be formed from the matrix.

For matrix A, it is denoted by ρ(A).

The rank of a matrix is said to be r if,

• There is at least one non-zero minor of order r.
• Every minor of matrix A having order higher than r is zero. 

Echelon form: A matrix is said to be in echelon form if

• Leading non-zero elements in each row is behind leading non-zero elements in the previous row.
• All the zero rows are below all the non-zero rows.

Steps to find the echelon form and rank of a matrix: 

• To reduce the matrix to the echelon form we can apply the Gauss elimination method on the matrix and can convert the matrix to an upper triangular matrix (lower off-diagonal elements zero).
• Then we can count the number of non -zero rows in this upper triangular matrix to get the rank of the matrix.

Calculation:

Let

 \(A = \left(\begin{matrix} 8 & 1 & 3 & 6 \\\ 0 & 3 & 2 & 2 \\\ -8 & -1 & -3 & -4 \end{matrix}\right)\)

by \(C_1 \rightarrow \frac{1}{8}C_1\)

\(A\ =\ \left(\begin{matrix} 1 & 1 & 3 & 6 \\\ 0 & 3 & 2 & 2 \\\ -1 & -1 & -3 & -4 \end{matrix}\right)\)

by \(R_3 \rightarrow R_3 + R_1\)

\(\Rightarrow \ A\ =\ \left(\begin{matrix} 1 & 1 & 3 & 6 \\\ 0 & 3 & 2 & 2 \\\ 0 & 0 & 0 & 2 \end{matrix}\right)\)

The last equivalent matrix is in the echelon form. The number of non-zero rows in this matrix is 3. Therefore it's ranked in 3.

Hence rank A = 3.

Important Points

Other properties of rank of a matrix are:

• The rank of a matrix does not change by elementary transformation, we can calculate the rank by changing the matrix into Echelon form. In the Echelon form, the rank of a matrix is the number of non-zero rows of the matrix.
• The rank of a matrix is zero if the matrix is null.
• ρ(A) ≤ min (Row, Column)
• ρ(AB) ≤ min [ρ(A), ρ(B)]
• ρ(AT A) = ρ(A AT) = ρ(A) = ρ(AT)
• If A and B are matrices of the same order, then ρ(A + B) ≤ ρ(A) + ρ(B) and ρ(A - B) ≥ ρ(A) - ρ(B).
• If Aθ is the conjugate transpose of A, then ρ(Aθ) = ρ(A) and ρ(A Aθ) = ρ(A).
• The rank of a skew-symmetric matrix cannot be one.