Show that every square matrix `A` can be uniquely expressed as `P+i Q ,w h e r ePa n dQ` are Hermitian matrices.

Prove that a sequence in an A.P., if the sum of its n terms is of the form A n^2+B n ,w h e r eA ,B are constants.

The number of terms in the expansion of (a+b+c)^n ,w h e r en in Ndot

Find the value of the determinant |b cc a a b p q r1 1 1|,w h e r ea ,b ,a n dc are respectively, the pth, qth, and rth terms of a harmonic progression.

Let Aa n dB be two nonsinular square matrices, A^T a n dB^T are the transpose matrices of Aa n dB , respectively, then which of the following are correct? B^T A B is symmetric matrix if A is symmetric B^T A B is symmetric matrix if B is symmetric B^T A B is skew-symmetric matrix for every matrix A B^T A B is skew-symmetric matrix if A is skew-symmetric

The position vectors of the vertices A ,B ,a n dC of a triangle are hat i+ hat j , hat j+ hat ka n d hat i+ hat k , respectively. Find the unite vector hat r lying in the plane of A B C and perpendicular to I A ,w h e r eI is the incentre of the triangle.

If the roots of the equation x^3+P x^2+Q x-19=0 are each one more that the roots of the equation x^3-A x^2+B x-C=0,w h e r eA ,B ,C ,P ,a n dQ are constants, then find the value of A+B+Cdot

If M is a 3xx3 matrix, where det M=1a n dM M^T=1,w h e r eI is an identity matrix, prove theat det (M-I)=0.

Show that a real value of x will satisfy the equation (1-i x)//(1+i x)=a-i b if a^2+b^2=1,w h e r ea ,b real.

Find the least value of n such that (n-2)x^2+8x+n+4>0,AAx in R ,w h e r en in Ndot

Given a matrix A=[a b c b c a c a b],w h e r ea ,b ,c are real positive numbers a b c=1a n dA^T A=I , then find the value of a^3+b^3+c^3dot