Functions Involving Parameters, Vectors, and Matrices

$\begin{pmatrix} 4 & -5 \\ 3 & 6 \end{pmatrix}$

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

$\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$

$\begin{pmatrix} -4 & 5 \\ -3 & -6 \end{pmatrix}$

Q has equal diagonal elements.

Q has symmetric elements across the diagonal.

Each row sum equals zero.

Each column sum equals zero.

Negative of the original number

The original number

Zero

One

Distributive Property

Commutative Property of Addition

Identity Property of Addition

Associative Property of Addition

The inverse of the vector

A zero vector

The original vector

A scalar multiple of the vector

A function that returns points to their original position.

A function that reflects points across the x-axis.

A function that rotates points 180 degrees counterclockwise.

A function that shifts points up by 90 units on the plane.

[[1,1],[1,1]]

[[1,0],[0,1]]

[[-1,0],[0,-1]]

[[0,0],[0,0]]

[0, 0]

[6, 10]

[3, 5]

[1, 1]

Scalar multiplication

Matrix inversion

Matrix subtraction

Determinant calculation

Each element on $C$'s main diagonal has to exceed unity after some power $k$ applied.

$C$'s eigenvalues should sum up to zero.

Trace($C$), summing up all diagonal elements, needs to match the number of dimensionality the space operates within.

$C$ must be orthogonal with determinant $\pm 1$.