Show that : The determinant of a matrix `A=([a_(i j)])_(5xx5)` where `a_(i j)+a_(j i)=0` for all `i` and `j` is zero.

Statement 1: The inverse of singular matrix A=([a_(i j)])_(nxxn), \ w h e r e \ a_(i j)=0,igeqj \ i s \ B=([a i j^-1])_(nxxn) . Statement 2: The inverse of singular square matrix does not exist.

construct 2xx2 matrix if A=[a_(ij)] whose elements a_(ij) are given by : (i-j)^2/2

If matrix A=[a_(ij)]_(3xx3) , matrix B=[b_(ij)]_(3xx3) , where a_(ij)+a_(ji)=0 and b_(ij)-b_(ji)=0 AA i , j , then A^(4)*B^(3) is

If a square matix a of order three is defined A=[a_("ij")] where a_("ij")=s g n(i-j) , then prove that A is skew-symmetric matrix.

Let A=([a_(i j)])_(3xx3) be a matrix such that AA^T=4Ia n da_(i j)+2c_(i j)=0,w h e r ec_(i j) is the cofactor of a_(i j)a n dI is the unit matrix of order 3. |a_(11)+4a_(12)a_(13)a_(21)a_(22)+4a_(23)a_(31)a_(32)a_(33)+4|+5lambda|a_(11)+1a_(12)a_(13)a_(21)a_(22)+1a_(23)a_(31)a_(32)a_(33)+1|=0 then the value of 10lambda is _______.

Let P=[a_(i j)] be a 3xx3 matrix and let Q=[b_(i j)],w h e r eb_(i j)=2^(i+j)a_(i j)for1lt=i ,jlt=3. If the determinant of P is 2, then the determinant of the matrix Q is 2^(10) b. 2^(11) c. 2^(12) d. 2^(13)

Show that the vectors are mutually perpendicular hat (i) + 2 hat (j) + hat (k) ,hat (i) + hat (j) - 3 hat (k) and 7 hat (i) - 4 hat (j) + hat (k)

Construct a 2xx2 matrix, where (i) a_("ij")=((i-2j)^(2))/(2) (ii) a_("ij")=|-2i+3j|

If projection of vector lambda hat (i) - hat (j) on vector hat(i) + hat(j) is zero , find lambda

Find the value of [ (hat (k)xx hat (j)).hat(i)+hat(j).hat (k)]