Question

Is it possible for a 5×5 matrix to be invertible when its columns do not span set of real numbers R^5​? Why or why​ not? Select the correct choice below.

A. It is not​ possible; according to the Invertible Matrix Theorem an n times nn×n matrix cannot be invertible when its columns do not span set of real numbers R^n.
B. It is​ possible; according to the Invertible Matrix Theorem an n times nn×n matrix can be invertible when its columns do not span set of real numbers R^n.
C. It will depend on the values in the matrix. According to the Invertible Matrix​ Theorem, a square matrix is only invertible if it is row equivalent to the identity.
D. It is​ possible; according to the Invertible Matrix Theorem all square matrices are always invertible.

Answer 1

The correct answer is A..

Explanation:

From the Invertible Matrix Theorem (IMT) we have a set of equivalent conditions to determine if a square matrix is invertible or not. In particular, it says that a square matrix of dimension tex]n\times n[/tex] is invertible if and only if, its columns span the vector space tex]R^n[/tex].

In the particular case of this exercise we have a matrix of dimension tex]5\times 5[/tex]. So, by the Invertible Matrix Theorem its columns must span the vector space tex]R^5[/tex]. Now, according to the statement of the exercise this condition does not hold. Hence, the given matrix cannot be invertible.