The rank of a square matrix equals its order (size) if and only if its determina
ID: 3110175 • Letter: T
Question
The rank of a square matrix equals its order (size) if and only if its determinate is nonzero. True or False: A homogeneous system of linear equations always has a solution. True or False: If .A and B? are invertible square matrices of the same size, then AB is also invertible, and (AD)^- 1 = A^- 1 B^- 1. True or False: Any vector space has a zero element. True or False: For a function of two variables f (x, y), the gradient nabla f is tangent to the level curve f (x, y) = 0. True or False: If two lines in the three - dimensional space don't intersect, then they are parallel. True or False: The value of the triple scalar product of three vectors on the order of the vectors. True or False: For differentiable vector functions f (t) and g (t), the Product Rule for Cross Products states that [f (t) times g(t)]' = F (t) times g (t) times g' (t) times f(t). True or False: n! gives the number of permutations of n distinguishable objects. True or False: If two events cannot happen simultaneously, they are independent. True or False: For a binomial distribution, mu = np. True or False: Sampling without replacement obeys the distribution law.Explanation / Answer
1. TRUE
THE RANK OF A SQUARE MATRIX EQUALS ITS ORDER IF AND ONLY IF ITS DETERMINANT IS NON-ZERO.
2.TRUE
A HOMOGENEOUS SYSTEM OF LINEAR EQUATION ALWAYS HAS A SOLUTION,.
3.FALSE
IF A AND B ARE INVERTIBLE MATRICES OF THE SAME SIZE THEN AB IS ALSO INVERTIBLE BUT (AB)^-1=(B^-1)(A^-1).
4.TRUE.
ANY VERCTOR SPACE HAS ZERO ELEMENT.
5. FALSE
FOR A FUNCTION OF TWO VARIABLES f(x,y) the gradient is tangent to the level curve f(x,y)=0.
6. FALSE
IF TWO LINES IN THREE DIMENSIONAL DONOT INTERSECT, THEN THEY ARE PARALLEL.
7. FLASE:
THE VALUE OF TRIPLE SCALE PRODUCT OF THREE VECTORS DEPENDS ON THE ORDER OF THE VECTORS.. (SINCE SCALAR PRODUCT IS COMMUTATIVE)
8. FALSE: FOR DIFFERENTIABLE VECTOR FUNCTIONS f(t) and g(t), the product for cross products staes that [f(t) x g(t)]'=f'(t)Xg(t) Xg'(t) X f(t).
9. TRUE:
n factorial gie the number of permutations of n distinguishable objects.
10. FALSE
If two events cannot happen simultabeously, they are indepenedent.
11. TRUE:
For a binomial distribution mean = n p.
12.FALSE:
Sampling without replacement obeys the hypergeometric distribution.