91Ó°ÊÓ

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

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

1. Ais an invertible matrix.
2. Ais row equivalent to the identity matrix of the \(n \times n\) 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.

The \(4 \times 4\) matrix \(\left[ {\begin{array}{*{20}{c}}1&3&7&4\\0&5&9&6\\0&0&2&8\\0&0&0&{10}\end{array}} \right]\) has four pivot positions. It is invertible according to part (c) of the invertible matrix theorem.

Thus, the matrix \(\left[ {\begin{array}{*{20}{c}}1&3&7&4\\0&5&9&6\\0&0&2&8\\0&0&0&{10}\end{array}} \right]\) is invertible.