For the matrix , find the numbers a and b such that A 2 + aA + bI = O .
Given matrix is,
A = [ 3 2 1 1 ]
The value of A 2 is,
A 2 = [ 3 2 1 1 ] [ 3 2 1 1 ] = [ 9 + 2 6 + 2 3 + 1 2 + 1 ] = [ 11 8 4 3 ]
Put the values of A and A 2 in A 2 + a A + b I = 0
[ 11 8 4 3 ] + a [ 3 2 1 1 ] + b [ 1 0 0 1 ] = [ 0 0 0 0 ] [ 11 8 4 3 ] + [ 3 a 2 a a a ] + [ b 0 0 b ] = [ 0 0 0 0 ] [ 11 + 3 a + b 8 + 2 a 4 + a 3 + a + b ] = [ 0 0 0 0 ]
Comparing both sides, we get,
3 a + b + 11 = 0 (1)
2 a + 8 = 0 (2)
a + 4 = 0 (3)
3 + a + b = 0 (4)
From equation (2)
2 a + 8 = 0 a = − 4
Substitute the value a in equation (4), we get,
− 4 + b + 3 = 0 b = 1
Thus a = − 4 and b = 1 .
Given matrix is,
The value of
Put the values of
Comparing both sides, we get,
From equation (2)
Substitute the value
Thus
,
find the numbers a and b such that A2
+ aA + bI = O.