Question

The determinant of all orthogonal 2 × 2 matrices is 1.

Short answer

False

Step-by-step solution

Step by step solutionStep 1: Orthogonal matrix

Inlinear algebra, an orthogonal matrix, is a realsquare matrixwhose columns and rows areorthonormalvectors.

One way to express this is${\mathit{Q}}^{\mathbf{T}}\mathit{Q}\mathbf{=}\mathit{Q}{\mathit{Q}}^{\mathbf{T}}\mathbf{=}\mathit{I}$where${\mathit{Q}}^{\mathbf{T}}$is thetransposeof$\mathit{Q}$and$\mathit{I}$is theidentity matrix.

Determine whether the statement is true or false

For example,

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ (identity transformation)
• $\det \left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]=1$

Hence, the statement is false.