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Add two matrices: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

$A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Subtract two matrices: $A = \begin{bmatrix} 9 & 8 \\ 7 & 6 \end{bmatrix}$ , $B = \begin{bmatrix} 5 & 4 \\ 3 & 2 \end{bmatrix}$

$A - B = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}$

Multiply a matrix by a scalar: $3 \times \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}$

$\begin{bmatrix} 6 & 3 \\ 0 & -3 \end{bmatrix}$

Matrix multiplication: $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ , $B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$

$A \times B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$

Transpose of a matrix: $A = \begin{bmatrix} 7 & 9 \\ 5 & 6 \end{bmatrix}$

$A^T = \begin{bmatrix} 7 & 5 \\ 9 & 6 \end{bmatrix}$

Determinant of a 2x2 matrix: $A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$

$\text{det}(A) = 4 \times 6 - 7 \times 2 = 10$

Inverse of a 2x2 matrix: $A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$

$A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}$

Element-wise multiplication (Hadamard product): $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

$A \circ B = \begin{bmatrix} 5 & 12 \\ 21 & 32 \end{bmatrix}$

Matrix-vector multiplication: $A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$ , $\textbf{v} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}$

$A \times \textbf{v} = \begin{bmatrix} 2 \times 5 + 4 \times 6 \\ 1 \times 5 + 3 \times 6 \end{bmatrix} = \begin{bmatrix} 34 \\ 23 \end{bmatrix}$

Matrix power: $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ , $A^{3}$

$A^{3} = A \times A \times A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix}$

Find the trace of a matrix: $A = \begin{bmatrix} 8 & 1 \\ 5 & 7 \end{bmatrix}$

$\text{trace}(A) = 8 + 7 = 15$

Rank of a matrix: $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

$\text{rank}(A) = 2$

Matrix norm: $\left\|A\right\|$ where $A = \begin{bmatrix} 3 & 4 \\ 0 & 5 \end{bmatrix}$ and using the Frobenius norm

$\left\|A\right\| = \sqrt{3^2 + 4^2 + 0^2 + 5^2} = \sqrt{9 + 16 + 0 + 25} = \sqrt{50}$

Eigenvalues of a matrix: $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$

Eigenvalues are $\lambda_1 = 2$ and $\lambda_2 = 3$

Diagonalize a matrix: $A = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}$

$P = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \; D = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}, \; P^{-1} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}, \; A = PDP^{-1}$

Cramer's Rule to solve $Ax = b$ for $x_1$ where $A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$ and $b = \begin{bmatrix} 4 \\ 11 \end{bmatrix}$

$x_1 = \frac{\text{det}(\begin{bmatrix} 4 & 1 \\ 11 & 3 \end{bmatrix})}{\text{det}(A)} = \frac{4 \cdot 3 - 1 \cdot 11}{2 \cdot 3 - 1 \cdot 5} = \frac{1}{1} = 1$

LU decomposition: $A = \begin{bmatrix} 4 & 3 \\ 6 & 3 \end{bmatrix}$

L = \begin{bmatrix} 1 & 0 \\ 1.5 & 1 \end{bmatrix}, U = \begin{bmatrix} 4 & 3 \\ 0 & -1.5 \end{bmatrix}, A = LU

Solve a system of linear equations using matrix inverse: $\begin{bmatrix} 3 & -1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \end{bmatrix}$

$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ 2 & 1 \end{bmatrix}^{-1} \begin{bmatrix} 5 \\ 7 \end{bmatrix} = \begin{bmatrix} 1/5 & 1/5 \\ -2/5 & 3/5 \end{bmatrix} \begin{bmatrix} 5 \\ 7 \end{bmatrix} = \begin{bmatrix} 2.4 \\ 2.6 \end{bmatrix}$

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