Program:
SIM
Course Code: 04170301
Exam Type: Midterm-A
Total points: 10 Points
Term and Year: Summer 2022
Course Title: Linear Algebra
Time Allowed: 40 Min.
Model Answer
Question 1 Consider
𝟏
𝑨 = [𝟎
𝟎
𝟎
√𝟐/𝟐
√𝟐/𝟐
𝟎
−√𝟐/𝟐 ] ,
√𝟐/𝟐
𝟏
𝒃 = [ √𝟐 ].
𝟑√𝟐
a) Show that the matrix 𝐀 is orthogonal . Then use this fact to evaluate 𝐝𝐞𝐭( 𝐀−𝟏 ) .
b) Use the orthogonality of 𝐀 to solve the system 𝐀𝐱 = 𝐛, where 𝐀 and 𝐛 are as given
above . Then determine 𝐓𝐫(𝑨).
1
Question 2
a) Use Gauss elimination method to find the flow pattern in the following network
assuming that the flows are all nonnegative.
A
80
𝒙𝟑
𝒙𝟏
𝒙𝟒
𝒙𝟐 B
20
C
2
b) Let 𝑨, 𝑩 and 𝑪 be 𝒏 × 𝒏 invertible matrices. Does the equation
𝑪−𝟏 (𝑨 + 𝑿)𝑩−𝟏 = 𝑰𝒏
have a solution 𝑿 ? If so, find it.
3