A matrix is a rectangular array of numbers arranged in rows and columns. To add two matrices, you add their corresponding elements. For example: \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \]

To multiply two matrices, take the dot product of rows from the first matrix and columns from the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix. For example: \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} \]

The determinant of a matrix is a scalar value that is computed from the elements of a square matrix and provides important properties about the matrix. For a 2x2 matrix: \[ \text{det}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc \] For example, the determinant of the matrix \( \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \) is \( (2 \times 4) - (3 \times 1) = 8 - 3 = 5 \).

An eigenvalue is a scalar \( \lambda \) such that when a matrix \( A \) is multiplied by a non-zero vector \( v \), the result is a scalar multiple of \( v \), i.e., \( A v = \lambda v \). The vector \( v \) is called an eigenvector corresponding to the eigenvalue \( \lambda \).

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] where \( \text{det}(A) \) is the determinant of the matrix \( A \). The inverse exists only if \( \text{det}(A) \neq 0 \).

A vector space is a collection of vectors that can be added together and multiplied by scalars, and that satisfies a set of axioms, including closure under addition and scalar multiplication, the existence of a zero vector, and distributivity of scalar multiplication with respect to vector addition and field addition.

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. If \( T \) is a linear transformation, then for any vectors \( u \) and \( v \) and any scalar \( c \), \( T(u + v) = T(u) + T(v) \) and \( T(cu) = cT(u) \).

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. To calculate the rank, perform row reduction (Gaussian elimination) to bring the matrix to row echelon form. The number of non-zero rows in the row echelon form is the rank of the matrix.

The dot product of two vectors \( \mathbf{a} = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \) is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]

To solve a system of linear equations \( A \mathbf{x} = \mathbf{b} \), where \( A \) is a matrix of coefficients, \( \mathbf{x} \) is a column vector of variables, and \( \mathbf{b} \) is a column vector of constants, you can use matrix inversion: \[ \mathbf{x} = A^{-1} \mathbf{b} \] This method works only if \( A \) is invertible, i.e., \( \text{det}(A) \neq 0 \).