Matrices and Vectors

• When a system of equations has more than two or three equations, it is difficult to discuss them without using matrices and vectors.
• A matrix is a rectangular array of numbers in which not only the value of the number is important but also its position in the array.
  
  $$A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{... ...y} \right]=\left[a_{ij}\right],~i=1,2,\ldots,n,~j=1,2,\ldots,m$$
  
  

• In general, $AB \neq BA$ ($B$ is another matrix), so the order of factors must be preserved in matrix multiplication.
• A matrix
  • with only one column, $n \times 1$ in size, is termed a column vector,
  • with only one row, $1 \times m$ in size, is called a row vector.
• When the term vector is used, it nearly always means a column vector.

• An $m \times n$ matrix times as $n \times 1$ vector gives an $m \times 1$ product ( $m \times n n \times 1$) .
• The general relation for $Ax=b$ is
  
  $$b_i=\sum_{k=1}^{No.of.cols.}a_{ik}x_{k},~~~~~~~i=1,2,\ldots,No.of.rows$$
  
  
  where $A$ is a matrix, $x$ and $b$ are vectors (column vectors).
• Set of linear equations
  
  $$\begin{array}{c} a_{11}x_1+a_{12}x_2+ \ldots +a_{1n}x_n =b_... ...\ a_{n1}x_1+a_{n2}x_2+ \ldots +a_{nn}x_n =b_n\\ \end{array}$$
  
  

• Much more simply in matrix notation, as $Ax=b$ where
  
  $$A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{... ...ay}{c} b_1\\ b_2\\ \vdots \\ b_n\\ \end{array}\right]$$
  
  

• Example,
  
  $$\left[ \begin{array}{rrr} 3 & 2 & 4 \\ 1 & -2 & 0 \\ -1... ...left[ \begin{array}{r} 14\\ -7\\ 2\\ \end{array}\right]$$
  
  

• is the same as the set of equations
  
  $$\begin{array}{l} 3x_1 + 2x_2 + 4x_3 = 14\\ x_1 - 2x_2= -7\\ -x_1 + 3x_2 + 2x_3 = 2\\ \end{array}$$