Rank of Matrix and Matrix Inverses
3 important questions on Rank of Matrix and Matrix Inverses
What is the rank of a matrix?
-If A is a mxn matrix over F, then rank(A) is the rank of the linear transfomation La: Fm->Fn
-if A is an nxn matrix over F, then A is invertible iff its rank is n
-rank a number, it represents the number of linearly independent rows of a matrix
-rank of a matrix is the dimension of the subspace generated by its columns.
-if A is a mxn matrix of rank r, r</=m, r</=n
-rank of Zero matrix is 0
- rank(At)=rank(A)
-if A is an nxn matrix over F, then A is invertible iff its rank is n
-rank a number, it represents the number of linearly independent rows of a matrix
-rank of a matrix is the dimension of the subspace generated by its columns.
-if A is a mxn matrix of rank r, r</=m, r</=n
-rank of Zero matrix is 0
- rank(At)=rank(A)
What are some rank preserving matrix properties?
-If A is mxn matrix, Pmxm and Qnxn are invertible, then
rank(AQ)=rank(A)
rank(PA)=rank(A)
rank(PAQ)=rank(A)
- Elementary row/column operations on a matrix are rank preserving
rank(AQ)=rank(A)
rank(PA)=rank(A)
rank(PAQ)=rank(A)
- Elementary row/column operations on a matrix are rank preserving
How to compute the inverse of a matrix?
-if Amxn is invertible, to find it's inverse we need to compute the augmented matrix (Amxn|Imxn), and reduce A to row echelon form while performing the same elementary row operations on Imxn.
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