Please state if each line below is true or false and provide an example. 1. True
ID: 2900853 • Letter: P
Question
Please state if each line below is true or false and provide an example.
1. True or false: A system comprising two linear equations in two variables has a unique solution if and only if the straight lines represented by the equations are nonparallel.
2. True or false: Suppose the straight lines represented by a system of two linear equations in two variables are parallel to each other. Then the system has infinitely many solutions.
3. True or false: If A and B are matrices of the same order, then .
4. True or false: If A and B are matrices such that AB and BA are both defined, then A and B must be square matrices.
5. True or false: If A is a square matrix with inverse and c is a nonzero real number, then .
6. True or false: If AX = B is a system of n linear equations in n unknowns and does not exist, then AX = B does not have a unique solution.
Explanation / Answer
1. True. If they are parallel then either they don't meet, or the equations represent the same line and there is an infinite number of solutions. If they aren't parallel, they intersect in a point.
2.True. Let the two lines be y1 = ax + b and y2 = ax + c
y1 = ax + b
y2 = ax + c
----------------- Subtract
y1 - y2 = b - c
If you now substitute for y1, you just get the second equation back, so there is no solution, which is pretty obvious because the lines do not intersect. The exception to this is when b and c are the same. Here the two parallel lines lie on top of each other and there is an infinite number of solutions
3. True(not clear)
4.True.
Lets assume that:
A is a 2 row x 3 column matrix
B is a 3 row x 2 column matrix
AB will result in a 2 x 2 matrix.
BA will result in a 3 x 3 matrix.
But AB + BA is not defined. For addition of matrices to be possible, the matrices have to be of the same size.
5.I believe the right side is (1/c) A^-1
if this is what you meant, then TRUE (not clear/if right side is what i assume then true)
6.
false