a) Show that the system of simultaneous linear equation
in the variables x1,x2,…, xn can be expressed as
AX = B, where A = [aij], X is an n × 1 matrix with xi the entry in its ith row, and B is an n × 1 matrix with bi the entry in its ith row.
b) Show that if the matrix A = [aij] is invertible (as defined in the preamble to Exercise 18), then the solution of the system in part (a) can be found using the equation X = A-1 B.
Solution:Step 1</p>
In this problem we need to show that a system of simultaneous linear equations
In variables can be written as
. We also need to show that the solution of the system of linear equations can be found using the equation
if the matrix
is invertible.
Step 2</p>
Consider a matrix A such that
and a
matrix X such that
.
The multiplication of the matrices A and X is possible since number of columns of A are equal to the number of rows of B so we write
If given simultaneous system of linear equations are
Then we can write
Hence it is shown that the given system of linear equations can be written in the form 
