how to find rank of a matrix and find M value shortcut
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Unfortunately, there is no shortcut to find the rank of matrix. We adapt different methods for different matrices based on various factors. For 2x2 and 3x3 we just check the linear dependency and determinant of sub-matrix.
For 3x3 matrix if determinant of matrix is non-zero then Rank = 3
else if, determinant of any one of the submatrix is non-zero then Rank = 2
else, the rank is 1 and the matrix is singular
Remember, Rank is zero only for null matrix.
NOT SATISFIED???
There is an alternative definition which provides a constructive approach to calculate the rank of a matrix, which is often adopted. Using elementary row transformations, the matrix is reduced to echelon form which has the following characteristics:
1. All zero rows are at the bottom of the matrix
2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
3. The leading entry in any nonzero row is 1.
4. All entries in the column above and below a leading 1 are zero.
===>The number of non-zero rows in the echelon form is equal to the rank of the matrix<===
For 3x3 matrix if determinant of matrix is non-zero then Rank = 3
else if, determinant of any one of the submatrix is non-zero then Rank = 2
else, the rank is 1 and the matrix is singular
Remember, Rank is zero only for null matrix.
NOT SATISFIED???
There is an alternative definition which provides a constructive approach to calculate the rank of a matrix, which is often adopted. Using elementary row transformations, the matrix is reduced to echelon form which has the following characteristics:
1. All zero rows are at the bottom of the matrix
2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
3. The leading entry in any nonzero row is 1.
4. All entries in the column above and below a leading 1 are zero.
===>The number of non-zero rows in the echelon form is equal to the rank of the matrix<===
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