Define Rank of Matrix with Example

what is Rank of Matrix?

The rank of Matrix A is equal to the number of nonzero rows in its echelon form.

or

The order of Ir in the canonical form of A.

define rank of matrix with example

To find the rank of any matrix we just reduce this matrix to its echelon form or canonical form and count its nonzero rows. p(A) shows the rank of matrix A.p(A) is the notation of the rank of a matrix. If a matrix has all elements are zero its rank is zero. If a matrix has not had full rank then it is said to be rank-deficient. The column rank and row rank are always equal. Moving towards firstly we know about the echelon form and canonical form.

A Matrix is in echelon form if it satisfies the following.

The first nonzero element in each row is 1.

Each leading entry is in a column to the right of the leading entry in the previous row.

A row echelon form is a canonical form. now we move to our main topic.

Example of the rank of a matrix

A=\begin{array}{c}\begin{array}{c}\begin{bmatrix}5&6&3\\1&0&8\\0&0&0\end{bmatrix}\end{array}\end{array}

Properties of the Rank of Matrix

  1. When the rank of a matrix is equal to its dimensions then it is said to be a full rank matrix.
  2. When the rank of the matrix is less than its original matrix then the answer is in the form of inconsistent (in variables form).
  3. If A has a square matrix then it is veritable if and only if it has full rank.
  4. The rank is at least one
  5. Only a zero matrix has zero ranks.
  6. The rank cannot be larger than the smallest dimension of the matrix.

How we find the rank of a matrix

We take a square matrix of order 3 by 3

\begin{array}{c}\begin{array}{c}\begin{bmatrix}5&9&3\\-3&5&6\\-1&-5&-3\end{bmatrix}\end{array}\end{array}

to find rank of matrix we reduce this matrix in an echelon form.

\begin{bmatrix}-1&-5&-3\\-3&5&6\\5&9&3\end{bmatrix}\; by R_{13}

\begin{bmatrix}1&5&3\\-3&5&6\\5&9&3\end{bmatrix}\; by(-1) R_1

\begin{bmatrix}1&5&3\\0&20&15\\0&-16&-12\end{bmatrix}\; by R_2 +3 R_1 and R_3 -5 R_1

\begin{bmatrix}1&5&3\\0&1&3/4\\0&-16&-12\end{bmatrix}\; by\frac1{20} R_2

\begin{bmatrix}1&5&3\\0&1&3/4\\0&0&0\end{bmatrix}\; by R_3 +16 R_2

this is the echelon form

at last, we count the number of non zero rows that is rank of matrix here the non zero rows are 2 so the rank of thus matrix is 2.

Why we Find the Rank of matrix?

The rank of the matrix tells us a lot about the matrix.

If we want to solve the system of linear equations we use the rank for this purpose. If the rank of any matrix equals the number of variables then we are able to find a unique solution of this matrix.

Note:
  1. When we reduce a matrix into its echelon form we can only elementary row transformations.
  2. the Nullity of a matrix n − r. where n = order of a matrix and r = rank of a matrix.
  3. rank of a matrix is zero only it case when the matrix has all entries are zero.

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