Question

If an \(n \times n\) matrix K cannot be row reduced to \({I_n}\), what can you say about the columns of K? Why?

Short answer

The columns of K are linearly dependent and do not span \({\mathbb{R}^n}\).

Step-by-step solution

Describe the given statement

Given that K cannot be row-reduced to \({I_n}\). Statement (b) of the invertible matrix theorem is false.

Use the inverse matrix theorem

The columns of K do not form a linearly independent set nor span \({\mathbb{R}^n}\). Since statement (b) is false, statements (e) and (h) are also false.

Draw a conclusion

Hence, the columns of K are linearly dependent and do not span\({\mathbb{R}^n}\).