Is it possible for a 55 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 nnn 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.

Question

Mathematics

Wyatt Marshall

18 November, 21:22

Is it possible for a 55 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 nnn 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.

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Answers (1)

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Cory Gilmore · Answer 1

The correct answer is A ...

Step-by-step 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.