If A=[1 −2
2 −1], B=[a 1
b −1] and (A + B)2 = A2 + B2, Find a and b.
Answers
Answer:
a =1 & b = 2
Step-by-step explanation:
A = 1 - 2
2 -1
A² = 1*1 + (-2*2) 1*(-2) + (-2)(-1)
2*1 + (-1)2 2*(-2) + (-1)(-1)
A² = -3 0
0 -3
B = a 1
b -1
B² = a²+b a-1
ba - b b+1
A² + B² = a² + b - 3 a-1
ba - b b-2
A+B = a +1 -1
a +b -2
(A+B)² = a² + 2a - b - 1 1 -a
ab + 2a - b - 2 2-b
(A+B)² = A² + B²
Equating each term
a² + b - 3 = a² + 2a - b - 1 => b = a + 1
a-1 = 1-a => 2a = 2 => a = 1
ba - b = ab + 2a - b - 2 => 2a = 2 => a = 1
b-2 = 2- b => 2b = 4 => b = 2
b = a + 1 is satisfied by a = 1 & b =2
(A+B)2 =
| (a+1)2-b+2 2-a-1 |
| (b+2)(a+1)-2b-4 4-b-2 |
A2 + B2 =
| a2+b-3 a-1 |
| ab-b. b-2 |
If (A+B)2 = A2 + B2 then 2a - a = a - 1 and b-2 = 4 - b - 2
Hence a = 1 & b = 2