Consider the following two statements with respect to the matrices Am $$\times$$ n , Bn $$\times$$ m , Cn$$\times$$ n and Dn $$\times$$ n .
Statement 1 : tr(AB) = tr(BA)
Statement 2 : tr(CD) = tr(DC)
where tr( ) represents the trace of a matrix. Which one of the following holds?
Which one of the following is the closed form for the generating function of the sequence (an}n $$\ge$$ 0 defined below?
$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$
Consider solving the following system of simultaneous equations using LU decomposition.
x1 + x2 $$-$$ 2x3 = 4
x1 + 3x2 $$-$$ x3 = 7
2x1 + x2 $$-$$ 5x3 = 7
where L and U are denoted as
$$L = \left( {\matrix{ {{L_{11}}} & 0 & 0 \cr {{L_{21}}} & {{L_{22}}} & 0 \cr {{L_{31}}} & {{L_{32}}} & {{L_{33}}} \cr } } \right),\,U = \left( {\matrix{ {{U_{11}}} & {{U_{12}}} & {{U_{13}}} \cr 0 & {{U_{22}}} & {{U_{23}}} \cr 0 & 0 & {{U_{33}}} \cr } } \right)$$
Which one of the following is the correct combination of values for L32, U32, and x1 ?
Which of the following is/are the eigen vector(s) for the matrix given below?
$$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {15} & 8 & 5 \cr {32} & {21} & 7 & {12} \cr } } \right)$$