Linear Systems and Matrices
An n by n linear system of equations is a system of n linear equations in n variables.
+ a12x2 + ... + a1nxn =
+ 3x2 = 9
To solve this we sequentially perform members of the following three operations:
Switching the two equations
Replace the 2nd equation with the 2nd equation + (-2)1st
Multiply the second equation by 1/7
Replace the 1st equation with the 1st equation + (2)2nd equation
An m by n matrix is an array of numbers with m rows and n columns.
The matrix below is a 2 by 3 matrix.
A square matrix is an n by n matrix, that is a matrix such that the number of rows is equal to the number of columns. The ijth entry is the number in the ith row and jth column. For example, the the matrix above the 1 2th entry is
a12 = 4
Note: A vector such at <2,4,6> can be looked as a 1 by 3 matrix.
A square matrix is called a diagonal matrix if
aij = 0 for i j
The matrix below is a diagonal matrix
If all the entries of a diagonal matrix are equal, then the matrix is called a scalar matrix. The example below is a scalar matrix.
Addition Subtraction and Scalar Multiplication
Just as with vectors we can add and subtract matrices and multiply a matrix by a scalar. To add or subtract matrices the dimensions of the two matrices must be the same.
Two matrices are called equal if all of their entries are equal.
If A is an m by n matrix, then the transpose of A, AT, is the n by m matrix with the rows and columns switched.
(AT)ij = Aji
In the above example
Example of the Theoretical Exercise
(AT)T = A
((AT)T )ij = (AT)ji = Aij
Since the ijth entries are equal for each ij, the matrices are equal.