STA3710 Assignment 4 (COMPLETE ANSWERS) 2025 (894289) - Due 9 September 2025

STA3710
ASSIGNMENT 4 2025

UNIQUE NO. 894289
DUE DATE: 9 SEPTEMBER 2025

, lOMoARcPSD|18222662




STA3710/014/0/2025


ASSIGNMENT 04
Unique Nr.: 894289
Fixed due date: 9 September 2025


Instructions

• Do not PLAGIARISE. Students suspected of plagiarism will be subjected to
disciplinary processes.
• Do not use any software to answer any of the questions. Only handwritten
answer sheets will be considered.
Question 1 [33]

1.1 Suppose that 𝐀 is an 𝑚 × 𝑚 matrix partitioned as

𝐀11 𝐀12
𝐀=( ),
𝐀 21 𝐀 22

where 𝐀11 is 𝑚1 × 𝑚1 and 𝑟𝑎𝑛𝑘(𝐀) = 𝑟𝑎𝑛𝑘(𝐀11 ) = 𝑚 1 .

Show that 𝐀 22 = 𝐀21 𝐀−1
11 𝐀12 . (5)

1.2 Consider the matrix
𝑎𝐈𝑚 𝑏𝐈𝑚
𝐀=( ),
𝑐𝐈𝑚 𝑑𝐈𝑚
where 𝑎, 𝑏, 𝑐 and 𝑑 are scalars. Given that 𝑚1 = 𝑚 2 and 𝐀11 𝐀21 = 𝐀 21 𝐀11 ,
prove that |𝐀| = |𝐀11 𝐀 22 − 𝐀 21 𝐀12 |. (5)
1.3 Let 𝐀 and 𝐁 be 𝑚 × 𝑛 and 𝑛 × 𝑚 matrices, respectively. Show that
|𝐈𝑚 + 𝐀𝐁| = |𝐈𝑛 + 𝐁𝐀|. (8)
1.4 Consider the positive semidefinite matrix
4 2 1
𝐀 = ( 2 4 2) .
1 2 4
𝐀 𝐀12
Suppose that 𝐀 can be partitioned as ( 11 ), where 𝐀11 is 2 × 2. Find the
𝐀 21 𝐀 22
Moore-Penrose inverse of 𝐀 (𝐀+). (8)
1.5 Find the determinant of
4 0 0 1 2
0 3 0 1 2
𝐀= 0 0 2 2 3 . (7)
0 0 1 2 3
(1 1 0 1 2)