1. Find a matrix p which transform matrix A [ a11=1,a12=0, a13= -1 a21=1, a22= 2, a23= 1 a31= 2, a32= 2 , a33=3] to diagonal form and hence calculate A^4 2. Using Cayley Hamilton theorem find the inverse and A^4. If A [ a11= 7, a12= 2, a13= -2 a21= -6, a22= -1 , a23= 2 a31= 6, a32= 2 , a33= 1] 3.Find rank, signature and index of the quadratic form 2x^2+y^2-3z^2+12xy-8yz by reducing it to canonical form . Also write the linear transformation which brings the normal reduction 4. find the orthogonal transformation which transform the quadratic form 6x^2+3y^2+3z^2-2yz into canonical form specify the nature of the quadratic form 5. (a) if lambda is an eigen value of a non singular matrix A prove that (1) 1/lambda is an eigen value of A inverse (2) |A|/lambda is an eigen value of adj A (3)lambda power n is an eigen value of A power n (b) prove that eigen values of a real symmetric matrix are real
Posted on: Sun, 20 Apr 2014 08:43:01 +0000
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