Matrices & Determinants¶
Introduction¶
| Name of Mathematician | Work in matrices |
|---|---|
| James Sylvester (1814-1897) | Used the word matrix first time |
| Arthur Cayley (1821-1895) | Used in linear transformations |
| Seki Kowa (1642-1708) | Concept of determinants |
| Gottfried Wilhelm Leibniz (1646-716) | Inventions of the determinants |
| Gabriel Crammer (1704-1752) | Used determinants for solving system of linear equations |
Matrix¶
- A rectangular array of numbers enclosed by a pair of brackets is called matrix.
OR
- Arrangement of numbers in rows and columns is called a matrix.
Rows¶
- The horizontal lines of numbers are called rows of matrix.
Columns¶
- The vertical lines of numbers are called columns of Matrix
Entries or Elements¶
- The numbers used in rows or columns are said to be the entries or elements of the matrix.
Order of a Matrix¶
- Order of matrix is defined as
\[ \text{Number of rows} \times \text{Number of columns} \]
General definition of a matrix¶
- Generally a brackets rectangular array of \(m \times n\) elements \(a_n\) (\(i=1,2,3,4...n\)), arranged in \(m\) rows and \(n\) columns is called an \(m\) by \(n\) matrix (written as \(m \times n\) matrix).
\[ \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} \]
Types of Matrices¶
Row matrix¶
A matrix which has only one row (ie \(1\times n\) matrix) is said to be a row matrix. For example:
\[ [1, 2, 3, 4] \]
Column matrix¶
A matrix which has only one column (ie an \(m\times 1\) matrix) is said to be a column matrix or column vector. For example:
\[ \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} \]
Square matrix¶
The matrix which has the same number of rows and columns, is called a square matrix.
OR
If \(m=n\) then the matrix of order \(m\times n\) is said yo be a square matrix of order \(n\) or \(m\). For example:
\[ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Rectangular matrix¶
A matrix, whose number of rows is not equal to number of rows is not equal to number of columns, is called a rectangular matrix.
OR
If \(m\not =\), then the matrix is called rectangular matrix of order \(m \times n\). For example:
\[ \begin{bmatrix} 2 & -1 & 7 \\ 9 & 5 & -1 \end{bmatrix} \]