Help Suppose U is an mm invertible matrix and A is an mn mat

Help: Suppose U is an m×m invertible matrix and A is an m×n matrix. Show that rank (UA)=rank (A).

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Solution

Given that U is a square matrix mxm and invertible

This implies U is non singular

Hence rank of U = No of rows in U = m

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Since A is mxn matrix the product UA is compatible and UA is having order mxn.

The product Ua, where a is any column vector of A, is a column vector lying in the column space of U.

Therefore, all columns of UA must be in the column space of U.

Or rank of UA = atmost rank of U.

Since U is having rank m,

rank of UA = rank of U = m