Question

Is it possible \({\bf{5}} \times {\bf{5}}\) matrix to be invertible when its columns do not span \({\mathbb{R}^{\bf{5}}}\)? Why or why not

Short answer

The inverse of the matrix does not exist.

Step-by-step solution

Consider the inverse of the matrix of order \({\bf{5}} \times {\bf{5}}\)

A matrix of the order \(5 \times 5\) has five columns that are linearly independent of each other. There is one vector that can be written as the linear combination of others.

Obtain the matrix of the order \({\bf{5}} \times {\bf{5}}\) after column reduction

As the column vectors in the given \(5 \times 5\) matrix are linearly independent, after column reduction, there will be one column with all entries as zero.

So, the determinant of the matrix is zero, and the inverse of the matrix does not exist.