Introduction

• A system of linear equations is strictly triangular if:
  • It is n x n, i.e. the number of equations is equal to the number of variables.
  • For equation k, the coefficients for x₁ to x_{k - 1} are 0.
  • For equation k, the coefficient for x_k is not 0.
  • For k in 1, ..., n
• A 3x3 strictly triangular system has the form:
• Or as an augmented matrix:

Solving strictly triangular systems

• Strictly triangular systems are very easy to solve.
• For example:
• From the 3rd equation, we know that:
• So we can substitute this into the second equation:
• And finally substitute x₂ and x₃ into our first equation:
• This method of solving strictly triangular systems is called back-substitution.

Converting systems to be strictly triangular

• If a system of equations has exactly one solution then it can be converted to a strictly triangular system.
• For example, given the system:
• First, we convert it into an augmented martrix:
• We start by eliminating the coefficients of x for all but the first row:
  • To do this, we subtract multiples of the 1st row from the other rows.
  • The row that is used for elimination is called the pivotal row.
  • We subtract the 1st row from the 2nd:
  • And then we subtract 3 times the 1st row from the 3rd row:
• Finally we eliminate the coefficients of y from the 3rd row:
  • To do this, we use the 2nd row as our pivotal row.
  • We subtract 3 times the 2nd row from the 3rd row:

Choosing a pivotal row

• Sometimes a system will be given like:
  • In this case the first row can't be used as a pivotal row to eliminate the coefficients in the first column.
  • Instead, it is necessary to switch round the first two rows.