6.2-03 Orthogonal matrices

Orthogonal matrices

Orthogonal matrices encode square changes of coordinates that preserve Euclidean geometry. The set links column tests with transpose inverses, row properties, determinant signs and calculations with rotations or reflections.

What you will practise

• Test square matrices for orthogonality.
• Solve systems using the transpose inverse.
• Check products and sign conventions.
• Derive consequences of norm preservation.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review matrix multiplication, transposes and inverses. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports isometries, QR factors and stable geometric transformations within Module 6 of the Complete Linear Algebra Course.