3.6 (f) Orthogonal Matrix :

1.1 Type of Matrices
1.2 Trace of a Matrix:
1.3 Transpose of a Matrix
1.4 Conjugate of a Matrix

2.1 (a) Addition and subtraction of Matrices:
2.2 (b) Multiplication of Matrices:

3.1 (a) Symmetric Matrix:
3.2 (b) Skew Symmetric Matrix:
3.3 (c) Hermitian matrix:
3.4 (d) Skew Hermitian Matrix:
3.5 (e) Singular and Non-singular Matrices:
3.6 (f) Orthogonal Matrix :
3.7 (g) Idempotent Matrix :
3.8 (h) Unitary Matrix :
3.9 (i) Involuntary Matrix:
3.10 (j) Nilpotent Matrix:

4.1 Minors & Cofactors
4.2 Properties of Determinant

5.1 Elementary row Transformations

6.1 Properties of Inverse of a Matrix
6.2 Elementary row Transformations

8.1 Echelon form of a Matrix
8.2 Rank of a Matrix

Any square matrix $A$ of order $n$ is said to be orthogonal if $AA' = A'A = {I_n}$.

Prove that $A = \frac{1}{3}\left[ {\begin{array}{c} 1&{ - 2}&2 \\ { - 2}&1&2 \\ { - 2}&{ - 2}&{ - 1} \end{array}} \right]$ is an orthogonal matrix.

Given $A = \frac{1}{3}\left[ {\begin{array}{c} 1&{ - 2}&2 \\ { - 2}&1&2 \\ { - 2}&{ - 2}&{ - 1} \end{array}} \right]$

$\therefore {A^T} = \frac{1}{3}\left[ {\begin{array}{c} 1&{ - 2}&{ - 2} \\ { - 2}&1&{ - 2} \\ { - 2}&{ - 2}&{ - 1} \end{array}} \right]$

$\therefore A{A^T} = \frac{1}{3}\left[ {\begin{array}{c} 1&{ - 2}&2 \\ { - 2}&1&2 \\ { - 2}&{ - 2}&{ - 1} \end{array}} \right] \times \frac{1}{3}\left[ {\begin{array}{c} 1&{ - 2}&{ - 2} \\ { - 2}&1&{ - 2} \\ { - 2}&{ - 2}&{ - 1} \end{array}} \right]$

$ = \frac{1}{9}\left[ {\begin{array}{c} {1 + 4 + 4}&{ - 2 - 2 + 4}&{ - 2 + 4 - 2} \\ { - 2 - 2 + 4}&{4 + 1 + 4}&{4 - 2 - 2} \\ { - 2 + 4 - 2}&{4 - 2 - 2}&{4 + 4 + 1} \end{array}} \right]$

$ = \frac{1}{9}\left[ {\begin{array}{c} 9&0&0 \\ 0&9&0 \\ 0&0&9 \end{array}} \right]$

$ = \left[ {\begin{array}{c} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$

$ = I$

Hence $A$ is an orthogonal matrix.