Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \right]$$.

Let $$\mathrm{det}(A)$$ and $$\mathrm{det}(B)$$ denote the determinates of the matrices A and B, respectively.

Which one of the options given below is TRUE?

Consider the following two statements with respect to the matrices A_{m $$\times$$ n ,} B_{n $$\times$$ m} , C_{n$$\times$$ 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 (a_n}_{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 ?