If A is an invertible matrix, then the columns of A^T are linearly independent. Explain Why.
Solution
Matrix A must have both rows and columns that are independent in order to be invertable. The transpose of A simple swaps rows for columns. Any non independent rows( or columns) in A will be non independent columns (or rows) in A^T.
Also, if you work out the values for the determinant of A, you will find that the determinant of A^T is the same. If A is invertable, the determinant does not equal zero, hence A^T determinate will also not equal zero and it too will be invertable.
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If A is an invertible matrix- then the columns of A^T are linearly ind.docx
1. If A is an invertible matrix, then the columns of A^T are linearly independent. Explain Why.
Solution
Matrix A must have both rows and columns that are independent in order to be invertable. The
transpose of A simple swaps rows for columns. Any non independent rows( or columns) in A
will be non independent columns (or rows) in A^T.
Also, if you work out the values for the determinant of A, you will find that the determinant of
A^T is the same. If A is invertable, the determinant does not equal zero, hence A^T determinate
will also not equal zero and it too will be invertable.
