Question

If all the columns of a square matrixAare unit vectors, then the determinant ofAmust be less than or equal to 1.

Short answer

Therefore, the columns of A are unit vectors, then it must apply $|\mathrm{de}tA|<1$and the given statement is true.

Step-by-step solution

Orthogonal Matrix Definition. 

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix.

Or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

To check whether the given condition is true or false. 

The determinant represents the volume of the solid which is formed by taking the column vectors as edges of the solid.

The maximum volume of such a solid will be when these vectors are orthogonal, which is 1.

The module of det A is the volume of the parallelepiped spanned by the columns of .

If all the columns of A are unit vectors, then it must apply$|\mathrm{de}tA|<1$.

Therefore, the given statement is true.