Let A be an nxn matrix. There are many statementsequivalent to the statement
Let A be an nxn matrix. There are many statements equivalent to the statement "A is invertible." "A has nonzero determinant" is one of them. Here are nine others: The system Ax = 0 has only the trivial solution. The column vectors of A are linearly independent. The row vectors of A are linearly independent. The column vectors of A span Rn. The row vectors of A span Rn. The column vectors of A form a basis for Rn. The row vectors of A form a basis for Rn. A has rank n. A has nullity 0. In the following nine short exercises, you will prove that these nine statements are all equivalent by showing that
