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 .