Functions Involving Parameters, Vectors, and Matrices

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.

[0, 0]

[6, 10]

[3, 5]

[1, 1]

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 $\pm 1$.

AB does not exist due to dimensional disagreement.

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

$\begin{pmatrix} b & c \end{pmatrix}$

$\begin{pmatrix} b & c \end{pmatrix}$

The original 2x2 matrix.

A scalar multiple of the original matrix.

A zero matrix.

An undefined result.

The zero matrix corresponding to the size of the original matrix

The original matrix multiplied by a two

The identity matrix corresponding to the size of the original matrix

A random matrix of equal dimensions

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

Distributive Property

Commutative Property of Multiplication

Associative Property of Multiplication

Inverse Property of Multiplication