Unless otherwise specified, assume that all matrices in these exercises are \(n \times n\). Determine which of the matrices in Exercises 1-10 are invertible. Use a few calculations as possible. Justify your answer.

10. [M] \[\left[ {\begin{array}{*{20}{c}}5&3&1&7&9\\6&4&2&8&{ - 8}\\7&5&3&{10}&9\\9&6&4&{ - 9}&{ - 5}\\8&5&2&{11}&4\end{array}} \right]\]

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.

Consider the matrix \(A = \left[ {\begin{array}{*{20}{c}}5&3&1&7&9\\6&4&2&8&{ - 8}\\7&5&3&{10}&9\\9&6&4&{ - 9}&{ - 5}\\8&5&2&{11}&4\end{array}} \right]\).

Use the code in MATLAB to obtain the row-reduced echelon form, as shown below:

\[\begin{array}{l} > > {\mathop{\rm A}\nolimits} \,\, = \,\,\left[ {5\,\,\,6\,\,\,7\,\,\,9\,\,\,8;\,3\,\,\,4\,\,\,5\,\,\,6\,\,\,5;\,1\,\,\,2\,\,\,3\,\,\,4\,\,\,2;\,\,7\,\,\,8\,\,\,10\,\,\, - 9\,\,\,11;\,\,9\,\,\, - 8\,\,\,9\,\,\, - 5\,\,\,4} \right]\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\]

\[\left[ {\begin{array}{*{20}{c}}5&3&1&7&9\\6&4&2&8&{ - 8}\\7&5&3&{10}&9\\9&6&4&{ - 9}&{ - 5}\\8&5&2&{11}&4\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{array}} \right]\]

Mark the non-zero leading entries in columns 1, 2, 3, and 4.

There are five pivot positions in the matrix.

The matrix has five pivot positions. Thus, it is invertible according to part (c) of the invertible matrix theorem.

Thus, the matrix is invertible.