91Ó°ÊÓ

Let Abe a square \(n \times n\) matrix. Then the following statements are equivalent.

For a given A, all these statements are either true or false.

1. Ais an invertible matrix.
2. Ais row equivalent to the \(n \times n\) matrix identity matrix.
3. Ahas n pivot positions.
4. The equation Ax = 0 has only a trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation \(x \mapsto Ax\) is one-to-one.
7. The equation \(Ax = b\) has at least one solution for each b in \({\mathbb{R}^n}\).
8. The columns of Aspan \({\mathbb{R}^n}\).
9. The linear transformation \(x \mapsto Ax\) maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).
10. There is an \(n \times n\) matrix Csuch that CA = I.
11. There is an \(n \times n\) matrix Dsuch that DA = I.
12. \({A^T}\) is an invertible matrix.

Interchange rows one and row two.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\{ - 4}&{ - 9}&7\end{aligned}} \right]\)

At row three, multiply row one by 4 and add it to row three.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\0&{ - 9}&{15}\end{aligned}} \right]\)

At row three, multiply row two by 3 and add it to row three.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\0&0&0\end{aligned}} \right]\)

According to part (b) of the invertible matrix theorem, the matrix is not row equivalent to the identity matrix.

Thus, the matrix \(\left[ {\begin{aligned}{*{20}{c}}0&3&{ - 5}\\1&0&2\\{ - 4}&{ - 9}&7\end{aligned}} \right]\) is not invertible.