What is a Matrix?
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, physics, engineering, and computer science to represent linear equations, transformations, and more.
Mathematically, a matrix \( A \) with \( m \) rows and \( n \) columns is written as:
\[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} \]
where \( a_{ij} \) represents the element at the \( i^{th} \) row and \( j^{th} \) column.
Plural Form of Matrix
The plural of “matrix” is matrices. This follows the Latin origin of the word where “-ix” endings change to “-ices” in the plural form.
Examples of Matrices
Example 1: A 2×2 Matrix
Consider a simple 2×2 matrix:
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]
This matrix has 2 rows and 2 columns.
Example 2: A 3×3 Matrix
Another common example is a 3×3 matrix:
\[ B = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 4 \\ 5 & -2 & 1 \end{bmatrix} \]
This matrix has 3 rows and 3 columns.
Example 3: Row and Column Matrices
A row matrix has only one row:
\[ R = \begin{bmatrix} 2 & 5 & 7 \end{bmatrix} \]
A column matrix has only one column:
\[ C = \begin{bmatrix} 3 \\ -1 \\ 4 \end{bmatrix} \]
Example 4: Zero Matrix
A matrix in which all elements are zero is called a zero matrix:
\[ Z = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]
Example 5: Identity Matrix
An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere:
\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]