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.
  
  $$A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{... ...y} \right]=\left[a_{ij}\right], i=1,2,\ldots,n, j=1,2,\ldots,m$$
  
  

• Two matrices of the same size may be added or subtracted. The sum of
  
  $$A=\left[a_{ij}\right],B=\left[b_{ij}\right]$$
  
  
  is the matrix whose elements are the sum of the corresponding elements of A and B.
  
  $$C=A+B=\left[a_{ij}+b_{ij}\right]=\left[c_{ij}\right]$$
  
  

• Similarly, we get the difference of two equal-sized matrices by subtracting corresponding elements. If two matrices are not equal in size, they cannot be added or subtracted.
• Multiplication of two matrices is defined as follows, when A is $n \times m$ and B is $m\times r$.
  
  $$\left[a_{ij}\right]*\left[b_{ij}\right]=\left[c_{ij}\right]=$$
  
  
  
  
  $$\left[ \begin{array}{ccc} (a_{11}b_{11}+a_{12}b_{21}+ \ldo... ... (a_{n1}b_{1r}+ \ldots +a_{nm}b_{mr})\\ \end{array} \right]$$
  
  
  
  
  $$c_{ij}=\sum_{k=1}^{m}a_{ik}b_{kj}, i=1,2,\ldots,n, j=1,2,\ldots,r$$
  
  

• Unless the number of columns of A equals the number of rows of B, the matrices cannot be multiplied.
• If A is $n \times m$, B must have is $m$ rows or else they are said to be nonconformable for multiplication and their product is undefined.
• In general, $AB \neq BA$, 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, and one of 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. The general relation for $Ax=b$ is
  
  $$b_i=\sum_{k=1}^{No.of.cols.}a_{ik}x_{k}, i=1,2,\ldots,No.of.rows$$
  
  
  This definition of matrix multiplication permits us to write the 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
  
  $$A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{... ...ay}{c} b_1\\ b_2\\ \vdots \\ b_n\\ \end{array}\right]$$
  
  
  For 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
  
  $$\begin{array}{l} 3x_1 + 2x_2 + 4x_3 = 14\\ x_1 - 2x_2= -7\\ -x_1 + 3x_2 + 2x_3 = 2\\ \end{array}$$
  
  

• A vector whose length is one is called a unit vector (The length of a vector is the square root of the sum of the squares of its components, an extension of the idea of the length of a two-component vector drawn from the origin).
• A vector that has all of its elements equal to zero is the zero vector. If all elements are zero except one, it is a unit basis vector. There are three distinct unit basis vectors of order-3:
  
  $$\left[ \begin{array}{c} 1\\ 0\\ 0\\ \end{array}\right... ... \left[ \begin{array}{c} 0\\ 0\\ 1\\ \end{array}\right]$$