Question

25. If the determinant of a square matrix is -1, then Amust be an orthogonal matrix.

Short answer

The given statement is false.

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

If the determinant of a square matrix is -1, thenmust be an orthogonal matrix or not.

For example,

$A=\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& 2\end{array}\right]$

$detA=\left[2\left(-\frac{1}{2}\right)-0\right]$

$detA=-1$

A is not an orthogonal matrix, yet det A = - 1.

Therefore, the given condition is false.