Linear Systems and Matrices Linear Systems An n by n linear system of equations is a system of n linear equations in n variables. a_{11}x_{1}
+ a_{12}x_{2} + ... + a_{1n}x_{n} =
b_{1}
Example Solve 2x_{1}
+ 3x_{2} = 9 Solution To solve this we sequentially perform members of the following three operations:
We have
Switching the two equations
Replace the 2nd equation with the 2nd equation + (2)1st
equation
Multiply the second equation by 1/7 Replace the 1st equation with the 1st equation + (2)2nd equation x_{1}
= 3 An m by n matrix is an array of numbers with m rows and n columns. Example 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 ij^{th} entry is the number in the i^{th} row and j^{th} column. For example, the the matrix above the 1 2^{th} entry is a_{12} = 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 a_{ij} = 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.
Example Let
then
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, A^{T}, is the n by m matrix with the rows and columns switched. (A^{T})_{ij} = A_{ji} In the above example
Example of the Theoretical Exercise Prove that (A^{T})^{T} = A
Solution We have ((A^{T})^{T} )_{ij} = (A^{T})ji = A_{ij} Since the ij^{th} entries are equal for each ij, the matrices are equal. Back to the Matrices and Applications Home Page Back to the Linear Algebra Home Page Back to the Math Department Home Page email Questions and Suggestions
