MAT1503 Assignment 3 Solutions 2026
UNISA
COVERS CHAPTERS 1 %2
DUE DATE: 31 JULY 2026 AT 09:00 PM
, Question 1
(a)
Given: A square matrix 𝑄is orthogonal if
𝑄 𝑇 = 𝑄 −1
or equivalently,
𝑄 𝑇 𝑄 = 𝐼.
We are required to prove that
det(𝑄) = ±1.
Proof
Since 𝑄is orthogonal,
𝑄 𝑇 𝑄 = 𝐼.
Take the determinant of both sides:
det(𝑄 𝑇 𝑄) = det(𝐼).
Using the determinant property
det(𝐴𝐵) = det(𝐴)det(𝐵),
we obtain
det(𝑄 𝑇 )det(𝑄) = 1.
Now use the property
det(𝑄 𝑇 ) = det(𝑄),
therefore
det(𝑄)det(𝑄) = 1.
Hence,
(det(𝑄))2 = 1.
UNISA
COVERS CHAPTERS 1 %2
DUE DATE: 31 JULY 2026 AT 09:00 PM
, Question 1
(a)
Given: A square matrix 𝑄is orthogonal if
𝑄 𝑇 = 𝑄 −1
or equivalently,
𝑄 𝑇 𝑄 = 𝐼.
We are required to prove that
det(𝑄) = ±1.
Proof
Since 𝑄is orthogonal,
𝑄 𝑇 𝑄 = 𝐼.
Take the determinant of both sides:
det(𝑄 𝑇 𝑄) = det(𝐼).
Using the determinant property
det(𝐴𝐵) = det(𝐴)det(𝐵),
we obtain
det(𝑄 𝑇 )det(𝑄) = 1.
Now use the property
det(𝑄 𝑇 ) = det(𝑄),
therefore
det(𝑄)det(𝑄) = 1.
Hence,
(det(𝑄))2 = 1.
