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Matrix Rank calculator finds the rank of given matrix, with detailed step by step calculations.
Examples to use Matrix Rank Calculator
1. Find the rank of the following 3×3 matrix.
Matrix \({\ A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} }\)
Solution
Since the given matrix is of size 3×3, follow these steps with the Matrix Rank Calculator.
- Enter matrix size of 3×3 in input fields: m=3 and n=3.
- A matrix with specified size of 3×3 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} \)
Row Echelon Form
Apply Row Transformations on the matrix A and reduce the matrix to Row Reduced Echelon Form.
In row-echelon form, the leading coefficient (pivot) of each nonzero row is always strictly to the right of the leading coefficient of the previous row.
Applying \( R2 → R2 - (4)R1 \)
\( = \begin{bmatrix}1 & 2 & 3\\0 & -3 & -6\\7 & 8 & 9\end{bmatrix} \)
Applying \( R3 → R3-(7)R1\)
\( = \begin{bmatrix}1 & 2 & 3\\0 & -3 & -6\\0 & -6 & -12\end{bmatrix} \)
Applying \( R1 → R1-(\frac{-2}{3})R2\)
\( = \begin{bmatrix}1 & 0 & -1\\0 & -3 & -6\\0 & -6 & -12\end{bmatrix} \)
Applying \( R3 → R3-(2)R2\)
\( = \begin{bmatrix}1 & 0 & -1\\0 & -3 & -6\\0 & 0 & 0\end{bmatrix} \)
This is the Row Echelon Form of the given matrix A.
Rank = Number of non-zero rows in the Row Echelon Form
All elements of \( R 1\) are not zero.
All elements of \( R 2\) are not zero.
All elements of \( R 3\) are zero.
There are 2 non-zero rows, which is the rank of the matrix.
Result
Therefore, the rank of given matrix A is
\(Rank~ of~ A = 2\)
2. Find the rank of the following 4×4 matrix.
Matrix \({\ A = \begin{bmatrix}1 & 2 & 3 & 6\\5 & 4 & 7 & 8\\9 & 6 & 3 & 2\\1 & 4 & 7 & 8\end{bmatrix} }\)
Solution
Since the given matrix is of size 4×4, follow these steps with the Matrix Determinant Calculator.
- Enter matrix size of 4×4 in input fields: m=4 and n=4.
- A matrix with specified size of 4×4 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 2 & 3 & 6\\5 & 4 & 7 & 8\\9 & 6 & 3 & 2\\1 & 4 & 7 & 8\end{bmatrix} \)
Row Echelon Form
Apply Row Transformations on the matrix A and reduce the matrix to Row Reduced Echelon Form.
In row-echelon form, the leading coefficient (pivot) of each nonzero row is always strictly to the right of the leading coefficient of the previous row.
Applying \( R2 → R2-(5)R1\)
\( = \begin{bmatrix}1 & 2 & 3 & 6\\0 & -6 & -8 & -22\\9 & 6 & 3 & 2\\1 & 4 & 7 & 8\end{bmatrix} \)
Applying \( R3 → R3-(9)R1\)
\( = \begin{bmatrix}1 & 2 & 3 & 6\\0 & -6 & -8 & -22\\0 & -12 & -24 & -52\\1 & 4 & 7 & 8\end{bmatrix} \)
Applying \( R4 → R4-(1)R1\)
\( = \begin{bmatrix}1 & 2 & 3 & 6\\0 & -6 & -8 & -22\\0 & -12 & -24 & -52\\0 & 2 & 4 & 2\end{bmatrix} \)
Applying \( R1 → R1-(\frac{-1}{3})R2\)
\( = \begin{bmatrix}1 & 0 & \frac{1}{3} & \frac{-4}{3}\\0 & -6 & -8 & -22\\0 & -12 & -24 & -52\\0 & 2 & 4 & 2\end{bmatrix} \)
Applying \( R3 → R3-(2)R2\)
\( = \begin{bmatrix}1 & 0 & \frac{1}{3} & \frac{-4}{3}\\0 & -6 & -8 & -22\\0 & 0 & -8 & -8\\0 & 2 & 4 & 2\end{bmatrix} \)
Applying \( R4 → R4-(\frac{-1}{3})R2\)
\( = \begin{bmatrix}1 & 0 & \frac{1}{3} & \frac{-4}{3}\\0 & -6 & -8 & -22\\0 & 0 & -8 & -8\\0 & 0 & \frac{4}{3} & \frac{-16}{3}\end{bmatrix} \)
Applying \( R1 → R1-(\frac{-1}{24})R3\)
\( = \begin{bmatrix}1 & 0 & 0 & \frac{-5}{3}\\0 & -6 & -8 & -22\\0 & 0 & -8 & -8\\0 & 0 & \frac{4}{3} & \frac{-16}{3}\end{bmatrix} \)
Applying \( R2 → R2-(1)R3\)
\( = \begin{bmatrix}1 & 0 & 0 & \frac{-5}{3}\\0 & -6 & 0 & -14\\0 & 0 & -8 & -8\\0 & 0 & \frac{4}{3} & \frac{-16}{3}\end{bmatrix} \)
Applying \( R4 → R4-(\frac{-1}{6})R3\)
\( = \begin{bmatrix}1 & 0 & 0 & \frac{-5}{3}\\0 & -6 & 0 & -14\\0 & 0 & -8 & -8\\0 & 0 & 0 & \frac{-20}{3}\end{bmatrix} \)
Applying \( R1 → R1-(\frac{1}{4})R4\)
\( = \begin{bmatrix}1 & 0 & 0 & 0\\0 & -6 & 0 & -14\\0 & 0 & -8 & -8\\0 & 0 & 0 & \frac{-20}{3}\end{bmatrix} \)
Applying \( R2 → R2-(\frac{21}{10})R4\)
\( = \begin{bmatrix}1 & 0 & 0 & 0\\0 & -6 & 0 & 0\\0 & 0 & -8 & -8\\0 & 0 & 0 & \frac{-20}{3}\end{bmatrix} \)
Applying \( R3 → R3-(\frac{6}{5})R4\)
\( = \begin{bmatrix}1 & 0 & 0 & 0\\0 & -6 & 0 & 0\\0 & 0 & -8 & 0\\0 & 0 & 0 & \frac{-20}{3}\end{bmatrix} \)
This is the Row Echelon Form of the given matrix A.
Rank = Number of non-zero rows in the Row Echelon Form
All elements of \( R 1\) are not zero.
All elements of \( R 2\) are not zero.
All elements of \( R 3\) are not zero.
All elements of \( R 4\) are not zero.
There are 4 non-zero rows, which is the rank of the matrix.
Result
Therefore, the rank of given matrix A is
\(Rank~ of~ A = 4\)