Matrices: Types and Operations

• A matrix constitutes a rectangular array of numbers or symbols organized in rows and columns.

General Form

• Matrices consist elements denoted as $a_{ij}$, where 'i' represents the row and 'j' represents the column.
• The size of a matrix is defined by the number of rows ($m$) and columns ($n$), expressed as $m \times n$.

Example Matrix

• The example provided is a $2 \times 3$ matrix with 2 rows and 3 columns.

Special Matrices

• Square Matrix: Has an equal number of rows and columns ($m = n$).
• Identity Matrix: A square matrix, represented by $I$, features 1s along its main diagonal and 0s in all other positions.
• Zero Matrix: All elements are 0.
• Diagonal Matrix: A square matrix where all non-diagonal elements are 0.
• Triangular Matrix: Either upper (elements below the diagonal are 0) or lower (elements above the diagonal are 0).

Matrix Operations

• Addition: Element-wise addition is possible only between matrices of identical sizes, resulting in a new matrix of the same size.
• Scalar Multiplication: Each element is multiplied by a scalar.
• Matrix Multiplication: The element $c_{ij}$ in the resulting matrix $C$ is the dot product of the i-th row of $A$ and the j-th column of $B$, provided $A$ is $m \times n$ and $B$ is $n \times p$, resulting in $C$ being $m \times p$.
• Matrix multiplication involves computing $c_{ij}$ as the sum of the products of $a_{ik}$ and $b_{kj}$ from $k = 1$ to $n$.

Transpose of a Matrix

• The transpose, denoted as $A^T$, swaps the rows and columns.
• If matrix A is $m \times n$, then $A^T$ is $n \times m$.

Determinant of a Matrix

• The determinant is a scalar value calculated from a square matrix.
• For a 2x2 matrix the determinant is calculated as $\det(A) = ad - bc$

Inverse of a Matrix

• Denoted as $A^{-1}$, satisfies $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.
• For a $2 \times 2$ matrix, the inverse is calculated using $A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$, given that $\det(A) = ad - bc \neq 0$.

Rank of a Matrix

• The rank is the maximum count of linearly independent rows (or columns).

Eigenvalues and Eigenvectors

• For a square matrix $A$, an eigenvector $v$ satisfies $Av = \lambda v$, where $\lambda$ is the eigenvalue.
• Eigenvalues are found by solving the characteristic equation $\det(A - \lambda I) = 0$.