If the matrices A and B are both non-singular then the inverse of the matrix AB is
Likewise, for any two conformable matrices A and B, we have the identity that the transpose of (AB) is
Whqat happens If the determinant of a 2 by 2 transformation matrix is negative
If the determinant is negative then the orientation of the shape has been reversed, such as what happens when the transformation is a reflection.
2 × 2 Transformation Matrices:
Reflection in the line y = x tan θ
Shear parallel to the x-axis, scale factor λ (x-axis invariant)
AND
Shear parallel to the y-axis, scale factor λ (y-axis invariant)
A shear moves every point by an amount that is proportional to its distance from the x- or y-axis. where λ is the constant of proportionality
3 × 3 Transformation Matrices - Reflection in the plane x = 0, y = 0, z = 0
arrangement of the xyz axes
what does the The determinant of a 3 × 3 matrix represent
Just as the determinant of a 2 × 2 transformation matrix gives the area scale factor of the transformation,
the determinant of a 3 × 3 transformation matrix gives the volume scale factor of the transformation.
Whqat happens If the determinant of a 3 by 3 transformation matrix is negative
A negative determinant indicates that the orientation of the image is reversed by the transformation, as in a
reflection
If X and Y are independent random variables then E(XY) = …
Var(aX + bY + c) = …
If you are given the probability distribution function of X, then how you can find the distribution function of related random variables, such as Y = X^2
multinomial expansion: (X1 +X2…..Xk)^n
