Algebra - Matrices

A Matrix is a rectangular array of numbers, symbols, or expressions arranged in horizontal rows and vertical columns. The order of a matrix is expressed as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns. Visually, a $2 \times 3$ matrix looks like a grid with 2 levels and 3 entries in each level.

A Square Matrix is a matrix where the number of rows equals the number of columns ($m = n$). A special type is the Identity Matrix ($I$), which is a square matrix with $1$s along the leading diagonal (from top-left to bottom-right) and $0$s everywhere else. It acts like the number '1' in matrix multiplication.

The Zero or Null Matrix ($O$) is a matrix in which all elements are zero. It can be of any order. Visually, it represents an empty state in matrix addition, where adding it to any matrix $A$ results in $A$ itself.

Two matrices are considered Equal if they have the exact same order and every element in the first matrix is identical to the corresponding element in the second matrix ($a_{ij} = b_{ij}$). This concept is frequently used to solve for unknown variables by setting up algebraic equations.

Matrix Addition and Subtraction can only be performed if the matrices have the same order. The operation is done by adding or subtracting corresponding elements. Visually, you can imagine placing one matrix over the other and combining the numbers that occupy the same position.

Scalar Multiplication involves multiplying every single element inside a matrix by a constant number (scalar) $k$. If you multiply a matrix by $-1$, you get its Additive Inverse, which, when added to the original matrix, results in a Null matrix.

Matrix Multiplication is possible only if the number of columns in the first matrix ($A$) equals the number of rows in the second matrix ($B$). If $A$ is $m \times n$ and $B$ is $n \times p$, the product $AB$ will have the order $m \times p$.

The Multiplication Process follows a 'Row-by-Column' rule. To find an element in the product, you move horizontally across a row of the first matrix and vertically down a column of the second matrix, multiplying corresponding pairs and summing the results. Note that matrix multiplication is generally not commutative ($AB \neq BA$).