What are covariant and contravariant vectors? Show that acceleration is a contravariant vector.

Q. A second order tensor has components ■(2&1&0@1&-3&-1@0&-1&2) in OX1X2X3. The transformation between the coordinates X1, X2, X3 and X1’ , X2’ , X3’ is defined by
X1’=1/3 (2x1+2x2-x3)
X2’=1/3 (2x1-x2+2x3)
X3’=1/3(-x1+2x2+2x3)
(i)Find its components in the new system OX1’X2’X3’.(By using Matrix Formula).
(ii) Compute A’32 in the new system.(By using component formula)

Q. Write transformation equations for a second order tensor Aij w.r.t this transformation and verify A’ii=Aii

WHAT IS MEAN BY (0,2) and (2,0) (1,1) tensor.

For the following matrix calculate the determinant:
{{5,3,7},{1,0,1},{9,6,4}}

Evaluate the following:
{{1,2,3},{4,5,6},{7,8,9}} * {{7,4},{6,4},{5,3}}

Given the following:
A = 3 * 3
B = 2 * 4
C = 4 * 3
D = 3 * 2
Which of the following are defined:
A * D
D * B
A * D^T
C * D + A * D^T
A * B
A * D^T + C

Evaluate the following problem:
2*{{3,5},{7,9}} + 3*{{7,6},{5,4}} - 4*{{9,8},{7,6}}

Solve the system of simultaneous equations below using any of the matrix method
2x + 3y - z = 6
3x + y + 5z =8
x + 2y + z =10

4x + 2y + 3z. = 35
x + 3y + 2z. = 45
2x + y + 5z. = 28
Calculat value of x,y & z
Crsmer's rule.