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Matrix RankCalculator

Matrix A size: x





Answer




How to use this Matrix Rank Calculator 🤔

Follow these steps to perform Matrix Rank for the given matrices.

  1. Enter the matrix size for matrix A.
  2. Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
  3. As and when you complete entering the matrix elements, Matrix Rank is calculated, and displayed in the answer section.

Matrix Rank

The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix. It is an important concept in linear algebra because it provides insights into the properties of the matrix, such as the dimension of the vector space spanned by its rows or columns.

The rank of a matrix A can be found by reducing the matrix to its row echelon form (REF) or reduced row echelon form (RREF) and counting the number of non-zero rows.

Example 1

Let's find the rank of the following matrix:

\( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)

Step 1: Convert A to its row echelon form (REF):

\( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix} \)

Step 2: Count the number of non-zero rows in the REF:

The matrix has 2 non-zero rows, so the rank of matrix A is 2.

Example 2

Consider another example with a different matrix:

\( A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} \)

Step 1: Convert A to its row echelon form (REF):

\( \begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Step 2: Count the number of non-zero rows in the REF:

The matrix has 1 non-zero row, so the rank of matrix A is 1.

These examples demonstrate the basic process of finding the rank of a matrix by reducing it to its row echelon form and counting the non-zero rows.