MAT2611 Assignment 8 2025 ANSWERS - Due 25 July 2025

MAT2611
ASSIGNMENT 8 2025
DUE 25 JULY 2025

, Instructions for the Assignments
Take care to explain all your arguments.
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MAT2611
ASSIGNMENT 08
Due date: Friday, 25 July 2025



Problem 1. Show the following:
(a) If A is an orthogonal matrix, then AT is also orthogonal matrix.
(b) If A and B are orthogonal matrices, then AB is also orthogonal matrix.




Problem 1

(a) If A is orthogonal, then AT is also orthogonal.
Definition: A matrix A is orthogonal if AT A = I (i.e. AT = A−1 ).
To show: AT is orthogonal, i.e. (AT )T AT = I.
But (AT )T = A, so:

(AT )T AT = AAT = I
Therefore, AT is also orthogonal.




(b) If A and B are orthogonal, then AB is also orthogonal.
Given: AT A = I and B T B = I
Check if (AB)T (AB) = I:

(AB)T (AB) = B T AT AB = B T IB = B T B = I
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Hence, AB is orthogonal.



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