Linear Algebra Answers

A parabola "y=ax^{2}+bx+c"  passes through the points (-2, 10); (1,4) and (2,6).

Which of the following are correct? More than one answer may be correct

1. The augmented matrix for the system of equations connecting a, b and c is "\\left(\\begin{array}{cccc}4&-2&1&:&10\\\\1&1&1&:&4\\\\4&2&1&: &6\\end{array}\\right)"
2. The determinant of the coefficient matrix for the system of equations connecting a, b and c is -12.
3. The parabola above passes through the origin.
4. By applying Gauss elimination, the normal form of the augmented matrix for the system of equations connecting a, b and c is"\\left(\\begin{array}{cccc}1&0&0&:&1\\\\0&1&0&:&-1\\\\0&0&1&: &4\\end{array}\\right).\n\n\u200b"

A system of linear equations has the augmented matrix

"\\left(\\begin{array}{cccc}1&1&k&:&1\\\\-1&1&2k&:&-3\\\\k&1&1&: &k+1\\end{array}\\right)"

where k

k is a constant. What is the value of k such that the system above has no solution?

1. "k=-1"
2. "k=-2"
3. "k=0"
4. "k=1"

Reduce the quadratic form

2 2 2 8 7 3 12 – 8 4 x y z xy yz zx    

to the canonical form

through an orthogonal transformation and hence show that it

is positive Semi-definite.

find the basis and the dimension of the solution space W of the following system

X1 + 2X2 - 2X3 + 2X4 - X5 = 0

X1 + 2X2 - X3 + 3X4 - 2X5 = 0

2X1 + 4X2 - 7X3 +X4 - X5 = 0