Examples:
(i)
Find AB & AC .
Note that AB = AC Þ B = C. However if B = C then AB = AC.
(ii) Find all matrices which commute with the matrix A=
[Ans:B
where x, y are scalars; Let A=
now equate AB = BA to get a = 0 and b = x.
(iii) R(t)=
Show that R(s) . R(t) = R(s+t)
(iv) If A=
and I is a unit matrix of order 2. show that
I + A = (I – A) .
[Solution : We have,
and
Now,
where t = tan (a/2)
(v) If A =
and B =
and (A + B)2 = A2 + B2. Find A and B.
[Ans. a = 1 , b = 4 ]
(vi) Let A =
and f (x) = x2 – 4x + 7, show that f (A) = 0.
Use this result to find A5 [Ans. A5 =
[Solution : We have , f (x) = x2 – 4x + 7 . Therefore f (A) = A2 – 4A + 7I2
Now, A2 =
f (A) = A2 – 4A + 7 I2
Now, f (A) = 0 Þ A2 -4A +7I2 =0
Þ A2 =4A-7I2
Þ A3=A2 A =(4A-7I2) A= 4A2 -7I2A
Þ A3 = 4(4A-7I2)-7A (Using A2 =4A -7I2)
Þ A3 =9A-28I2
Þ A3 9A -28I2
Þ A4 A3A=(9A-29I2)A
Þ A4 =9A2-28A= 9(4A-7I2 - 28A(Using A2 =4A -7I2 )
ÞA4 = 36A -63I2 -28A = 8A -63I2
ÞA5 = A4A = (8A-63I2 ) A = 8A2 - 63I2A
ÞA5 = 8 (4A -7I2 ) – 63A =- 31A – 56I2 (Using A2 -4A -7I2)
(vii) Find an upper triangular matrix A such that ;
Let A =
and proceed and note that for an upper triangular matrix .aij =0 " i > j
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