### Matrix & Vector Properties

### What is Rank of a Matrix?

Rank of a matrix is the number of non-zero rows in the Reduced Row Eschelon Form of the matrix.

### What is Trace of a Matrix?

Trace of a matrix is calculated by summing up the elements on the main diagonal of the matrix. $$\mathrm{tr}(A_{n \times n}) = \sum_{i}^{n} a_{ii}$$

### Characteristic Polynomial of a Matrix

Characteristic polynomial of a square matrix \(A_{n \times n}\), denoted by \(p_A(\lambda)\) is defined as: $$p_A(\lambda) = \det(A - \lambda I)$$

### What is Inverse Matrix?

Inverted Matrix (or Inverse) of a square matrix \(A_{n \times n}\) is a matrix denoted by \(A^{-1}\) satisfies $$AA^{-1} = A^{-1}A = I_{n}$$ where \(I_{n}\) is the n-by-n identity matrix. Inverse matrix of \(A_{n \times n}\) is defined as: $$A^{-1} = \frac{1}{\det(A)}\mathrm{adj}(A)$$ if \(\det(A) \neq 0\).

### What is Matrix of Minors?

Minor \(M_{ij}\) of a square matrix \(A_{n \times n}\) is defined as the determinant value of the resulting matrix after crossing out the row i and column j from the original matrix A. $$M = \begin{pmatrix}M_{11} & \cdots & M_{1n}\\ \vdots & \ddots & \vdots \\ M_{n1} & \cdots & M_{nn} \end{pmatrix}$$

### What is Matrix of Cofactors?

Matrix of Cofactors (or Cofactor Matrix) of a square matrix \(A_{n \times n}\) is defined as: $$C = \begin{pmatrix}C_{11} & \cdots & C_{1n}\\ \vdots & \ddots & \vdots \\ C_{n1} & \cdots & C_{nn} \end{pmatrix}$$ where \(C_{ij} = (-1)^{(i + j)}M_{ij} \quad (1 \leq i, \quad j \leq n)\) where \(M_{ij}\) is the minors of A.

### What is Adjugate (Adjoint) Matrix?

Adjugate (or Adjoint) of a matrix \(A_{n \times n}\) is defined as the transpose of the cofactor matrix of \(A_{n \times n}\), thus \( \mathrm{adj}(A) = C^T \).

### Unit Vector Definition

Unit vector is a vector which has a magnitude of one. Unit vector of a vector is: \(\hat{v} = \frac{\overrightarrow{v}}{|v|}\).

### Magnitude of a Vector

Magnitude (length) of a vector is defined as: \(\Vert \overrightarrow{v} \Vert = \sqrt{\sum_{i}^{n} v_i^2}\).