MAT2611 Assignment 08 2025 (Accurate Solutions) Due Friday, 25 July 2025

MAT2611 Assignment 08 Solutions
Due date: Friday, 25 July 2025

Problem 1
Problem Statement: Show the following:
(a) If A is an orthogonal matrix, then AT is also an orthogonal matrix.

(b) If A and B are orthogonal matrices, then AB is also an orthogonal
matrix.

Part (a)
Step 1: An orthogonal matrix satisfies AT A = I.
Step 2: Compute (AT )T AT = AAT = I.
Step 3: Compute AT (AT )T = AT A = I.
Step 4: Since both conditions hold, AT is orthogonal.
Final Answer: AT is an orthogonal matrix.

Part (b)
Step 1: Given AT A = I, B T B = I.
Step 2: Compute (AB)T (AB) = B T AT AB = B T IB = I.
Step 3: Compute (AB)(AB)T = ABB T AT = AIAT = I.
Step 4: Since both conditions hold, AB is orthogonal.
Final Answer: AB is an orthogonal matrix.

Problem 2
Problem Statement: Determine which of the following matrices are or-
thogonal. For those that are, find the inverse.
" #
√1 √1
2 2
(a) √1
2
− √12
 
1 0 0
(b)  √12 0 √1 
2
0 1 0

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