Matrices & Determinants
Orthogonal matrix
Grade Class 12
Question:
Which of the following is an orthogonal matrix -<br><img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%202/7%20%26%203/7%20%5C%5C%202/7%20%26%203/7%20%26%206/7%20%5C%5C%203/7%20%26%20-6/7%20%26%202/7%20%5Cend%7Bpmatrix%7D" alt="(A)"><br><img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%202/7%20%26%203/7%20%5C%5C%202/7%20%26%20-3/7%20%26%206/7%20%5C%5C%203/7%20%26%206/7%20%26%20-2/7%20%5Cend%7Bpmatrix%7D" alt="(B)"><br><img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%20-6/7%20%26%20-2/7%20%26%20-3/7%20%5C%5C%202/7%20%26%203/7%20%26%206/7%20%5C%5C%20-3/7%20%26%206/7%20%26%202/7%20%5Cend%7Bpmatrix%7D" alt="(C)"><br><img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%20-2/7%20%26%203/7%20%5C%5C%202/7%20%26%202/7%20%26%20-3/7%20%5C%5C%20-6/7%20%26%202/7%20%26%203/7%20%5Cend%7Bpmatrix%7D" alt="(D)">
(A) <img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%202/7%20%26%203/7%20%5C%5C%202/7%20%26%203/7%20%26%206/7%20%5C%5C%203/7%20%26%20-6/7%20%26%202/7%20%5Cend%7Bpmatrix%7D">
(B) <img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%202/7%20%26%203/7%20%5C%5C%202/7%20%26%20-3/7%20%26%206/7%20%5C%5C%203/7%20%26%206/7%20%26%20-2/7%20%5Cend%7Bpmatrix%7D">
(C) <img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%20-6/7%20%26%20-2/7%20%26%20-3/7%20%5C%5C%202/7%20%26%203/7%20%26%206/7%20%5C%5C%20-3/7%20%26%206/7%20%26%202/7%20%5Cend%7Bpmatrix%7D">
(D) <img src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%206/7%20%26%20-2/7%20%26%203/7%20%5C%5C%202/7%20%26%202/7%20%26%20-3/7%20%5C%5C%20-6/7%20%26%202/7%20%26%203/7%20%5Cend%7Bpmatrix%7D">
Step-by-Step Solution
Key Concept: A matrix A is orthogonal if A*A^T = I, which means the sum of squares of elements in each row/column is 1 and the dot product of any two distinct rows/columns is 0.
For a matrix to be orthogonal, the sum of squares of elements in each row must be 1. For option (A): (6/7)^2 + (2/7)^2 + (3/7)^2 = (36+4+9)/49 = 49/49 = 1. Checking dot product of row 1 and row 2: (6/7)*(2/7) + (2/7)*(3/7) + (3/7)*(-6/7) = (12+6-18)/49 = 0. Thus, (A) is an orthogonal matrix.
Correct Answer: 1