Question 6
Find the inverse of the matrix [■8(−3&2@5&−3)]. Hence, find the matrix P satisfying the matrix equation P [■8(−3&2@5&−3)] = [■8(1&2@2&−1)].
Let A = [■8(−3&2@5&−3)]
We know that
A-1 = 1/(|A|) adj A
exists when |A|≠ 0
|A| = |■8(−3&2@5&−3)|
= (–3) × (–3) – 5 × 2 = 9 – 10 = –1
Since |A|≠ 0, A–1 exists
Calculating adj A
A =[■8(−3&2@5&−3)]
adj(A) = [■8(−3&2@5&−3)]
= [■8(−3&−2@−5&−3)]
Calculating A–1
A-1 = 1/(|A|) adj A
= 𝟏/(−𝟏) [■8(−3&−2@−5&−3)]
= −1[■8(−3&−2@−5&−3)] = [■8(−1 × −3&−1 × −2@−1 × −5&−1 × −3)]
= [■8(𝟑&𝟐@𝟓&𝟑)]
Now,
P [■8(−3&2@5&−3)] = [■8(1&2@2&−1)]
PA = [■8(1&2@2&−1)]
P = [■8(1&2@2&−1)]A–1
Putting value of A–1
P = [■8(1&2@2&−1)][■8(3&2@5&3)]
P = [■8(1(3)+2(5)&1(2)+2(3)@2(3)+(−1)5&2(2)+(−1)3)]
P = [■8(3+10&2+6@6−5&4−3)]
P = [■8(𝟏𝟑&𝟖@𝟏&𝟏)]

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.