MAT2611 ASSIGNMENT 8 2025
Problem 1
(a)
Let 𝐴 be an 𝑛 × 𝑛 matrix.
Suppose that 𝐴 is orthogonal
⇒ 𝐴𝐴𝑇 = 𝐼𝑛
Note that orthogonal matrices are invertible
⇒ 𝐴−1 𝐴𝐴𝑇 = 𝐴−1 𝐼𝑛
⇒ 𝐴𝑇 = 𝐴−1
⇒ (𝐴𝑇 )−1 = ( 𝐴−1 )−1
⇒ (𝐴𝑇 )−1 = 𝐴
⇒ 𝐴 = (𝐴𝑇 )−1
⇒ (𝐴𝑇 )𝐴 = (𝐴𝑇 )(𝐴𝑇 )−1
⇒ 𝐴𝑇 𝐴 = 𝐼𝑛
Note that 𝐴 = (𝐴𝑇 )𝑇
⇒ (𝐴𝑇 )(𝐴𝑇 )𝑇 = 𝐼𝑛
⇒ 𝐴𝑇 is orthogonal
(b)
Suppose that 𝐴 and 𝐵 are 𝑛 × 𝑛 matrices that are both orthogonal
⇒ 𝐴𝐴𝑇 = 𝐼𝑛 and 𝐵𝐵 𝑇 = 𝐼𝑛
⇒ 𝐴𝑇 = 𝐴−1 and 𝐵 𝑇 = 𝐵 −1
⇒ 𝐵 𝑇 = 𝐵 −1 and 𝐴𝑇 = 𝐴−1
⇒ 𝐵 𝑇 × 𝐴𝑇 = 𝐵 −1 × 𝐴−1
⇒ 𝐵 𝑇 𝐴𝑇 = 𝐵 −1 𝐴−1
Problem 1
(a)
Let 𝐴 be an 𝑛 × 𝑛 matrix.
Suppose that 𝐴 is orthogonal
⇒ 𝐴𝐴𝑇 = 𝐼𝑛
Note that orthogonal matrices are invertible
⇒ 𝐴−1 𝐴𝐴𝑇 = 𝐴−1 𝐼𝑛
⇒ 𝐴𝑇 = 𝐴−1
⇒ (𝐴𝑇 )−1 = ( 𝐴−1 )−1
⇒ (𝐴𝑇 )−1 = 𝐴
⇒ 𝐴 = (𝐴𝑇 )−1
⇒ (𝐴𝑇 )𝐴 = (𝐴𝑇 )(𝐴𝑇 )−1
⇒ 𝐴𝑇 𝐴 = 𝐼𝑛
Note that 𝐴 = (𝐴𝑇 )𝑇
⇒ (𝐴𝑇 )(𝐴𝑇 )𝑇 = 𝐼𝑛
⇒ 𝐴𝑇 is orthogonal
(b)
Suppose that 𝐴 and 𝐵 are 𝑛 × 𝑛 matrices that are both orthogonal
⇒ 𝐴𝐴𝑇 = 𝐼𝑛 and 𝐵𝐵 𝑇 = 𝐼𝑛
⇒ 𝐴𝑇 = 𝐴−1 and 𝐵 𝑇 = 𝐵 −1
⇒ 𝐵 𝑇 = 𝐵 −1 and 𝐴𝑇 = 𝐴−1
⇒ 𝐵 𝑇 × 𝐴𝑇 = 𝐵 −1 × 𝐴−1
⇒ 𝐵 𝑇 𝐴𝑇 = 𝐵 −1 𝐴−1
