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Matrix Operations

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2x2 matrix addition

egin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

Scalar multiplication of a 3x3 matrix

2 \cdot \begin{bmatrix} 1 & 0 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

\begin{bmatrix} 2 & 0 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \end{bmatrix}

Matrix subtraction 2x2

\begin{bmatrix} 10 & 8 \\ 6 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

\begin{bmatrix} 9 & 6 \\ 3 & 0 \end{bmatrix}

Transposing a 3x2 matrix

\left( \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \right)^T

\begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}

2x2 matrix multiplication

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Determinant of a 2x2 matrix

det\left( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \right)

-2

Determinant of a 3x3 matrix

det\left( \begin{bmatrix} 6 & 1 & 1 \\ 4 & -2 & 5 \\ 2 & 8 & 7 \end{bmatrix} \right)

-306

Inverse of a 2x2 matrix

\left( \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \right)^{-1}

\begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}

Matrix exponentiation (squared) of a 2x2 matrix

\left( \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \right)^2

\begin{bmatrix} 7 & 18 \\ 6 & 19 \end{bmatrix}

Rank of a matrix

Rank\left( \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix} \right)

The trace of a 3x3 matrix

Trace\left( \begin{bmatrix} 9 & 0 & 1 \\ 0 & 7 & 2 \\ 1 & 0 & 4 \end{bmatrix} \right)

Eigenvalues of a 2x2 matrix

Eigenvalues\left( \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \right)

2, 4

Matrix-vector multiplication

\begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix}

\begin{bmatrix} 10 \\ 4 \end{bmatrix}

Outer product of two vectors

\begin{bmatrix} 2 \\ 3 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 4 \end{bmatrix}

\begin{bmatrix} 2 & 8 \\ 3 & 12 \end{bmatrix}

Tensor product of two 2x2 matrices

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \otimes \begin{bmatrix} 0 & 5 \\ 6 & 7 \end{bmatrix}

\begin{bmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{bmatrix}

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