Ex 4.5, 14 - Chapter 4 Class 12 Determinants (Term 1)

Transcript

Ex 4.5, 14 For the matrix A = [■8(3&[email protected]&1)] , find the numbers a and b such that A2 + aA + bI = O. Finding A2 A2 = A.A = [■8(3&[email protected]&1)] [■8(3&[email protected]&1)] = [■8(3(3)+2(1)&3(2)+2(1)@1(3)+1(1)&1(2)+1(1))] = [■8(9+2&[email protected]+1&2+1)] = [■8(11&[email protected]&3)] Now, A2 + aA + bI = O Putting values [■8(11&[email protected]&3)] + a [■8(3&[email protected]&1)] + b [■8(1&[email protected]&1)] = O [■8(11&[email protected]&3)] + [■8(3a&[email protected]&a)] + [■8(b&[email protected]&b)] = O [■8(11+3a+b&[email protected]+a+0&3+a+b)] = O [■8(3a+b+11&[email protected]+a&a+b+3)] = [■8(0&[email protected]&0)] Since the matrices are equal, Comparing corresponding elements 3a + b + 11 = 0 2a + 8 = 0 4 + a = 0 a + b + 3 = 0 Solving (3) a + 4 = 0 a = –4 Putting value a in (1) 11 + 3 a + b = 0 11 + 3 (–4) + b = 0 11 – 12 +b = 0 –1 + b = 0 …(1) b = 1 Hence, a = −4, b = 1