1
GATE CSE 2023
+1
-0.33

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?

A
$$\mathrm{det}(A)=\mathrm{det}(B)$$
B
$$\mathrm{det}(B)=-\mathrm{det}(A)$$
C
$$\mathrm{det}(A)=0$$
D
$$\mathrm{det}(AB)=\mathrm{det}(A)+\mathrm{det}(B)$$
2
GATE CSE 2022
+1
-0.33

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?

A
Statement 1 is correct and Statement 2 is wrong.
B
Statement 1 is wrong and Statement 2 is correct.
C
Both Statement 1 and Statement 2 are correct.
D
Both Statement 1 and Statement 2 are wrong.
3
GATE CSE 2022
+1
-0.33

Which one of the following is the closed form for the generating function of the sequence (an}n $$\ge$$ 0 defined below?

$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$

A
$${{x(1 + {x^2})} \over {{{(1 - {x^2})}^2}}} + {1 \over {1 - x}}$$
B
$${{x(3 - {x^2})} \over {{{(1 - {x^2})}^2}}} + {1 \over {1 - x}}$$
C
$${{2x} \over {{{(1 - {x^2})}^2}}} + {1 \over {1 - x}}$$
D
$${x \over {{{(1 - {x^2})}^2}}} + {1 \over {1 - x}}$$
4
GATE CSE 2022
+1
-0.33

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 ?

A
L32 = 2, U33 = $$- {1 \over 2}$$, x1 = $$-$$ 1
B
L32 = 2, U33 = 2, x1 = $$- {1 \over 2}$$
C
L32 = $$- {1 \over 2}$$, U33 = 2, x1 = 0
D
L32 = $$- {1 \over 2}$$, U33 = $$- {1 \over 2}$$, x1 = 0
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