1
GATE CSE 2022
MCQ (Single Correct Answer)
+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
2
GATE CSE 2022
MCQ (More than One Correct Answer)
+1
-0.33

Which of the following is/are the eigen vector(s) for the matrix given below?

$$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {15} & 8 & 5 \cr {32} & {21} & 7 & {12} \cr } } \right)$$

A
$$\left( {\matrix{ { - 1} \cr 1 \cr 0 \cr 1 \cr } } \right)$$
B
$$\left( {\matrix{ 1 \cr 0 \cr { - 1} \cr 0 \cr } } \right)$$
C
$$\left( {\matrix{ { - 1} \cr 0 \cr 2 \cr 2 \cr } } \right)$$
D
$$\left( {\matrix{ 0 \cr 1 \cr { - 3} \cr 0 \cr } } \right)$$
3
GATE CSE 2019
MCQ (Single Correct Answer)
+1
-0.33
Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?
A
I implies II; II does not imply I
B
II implies I; I does not imply II
C
I does not imply II; II does not imply I
D
I and II are equivalent statements
4
GATE CSE 2018
Numerical
+1
-0
Consider a matrix $$A = u{v^T}$$ where $$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ 1 \cr 1 \cr } } \right).$$ Note that $${v^T}$$ denotes the transpose of $$v.$$ The largest eigenvalue of $$A$$ is _____.
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