Upper triangular form

Definition

An upper triangular form of a matrix is one where all the entries below the main diagonal are zero. This form is used to simplify solving systems of linear equations.

5 Must Know Facts For Your Next Test

1. In an upper triangular matrix, all elements below the main diagonal are zero.
2. Upper triangular form is useful for back substitution in solving linear systems.
3. The determinant of an upper triangular matrix is the product of its diagonal elements.
4. Row operations can transform a matrix into upper triangular form.
5. Gauss-Jordan elimination often involves converting a system's augmented matrix to upper triangular form.

Review Questions

• What does it mean for a matrix to be in upper triangular form?
• How does back substitution work with an upper triangular matrix?
• Why is the determinant of an upper triangular matrix simply the product of its diagonal elements?

Related terms

Gaussian Elimination: A method for solving systems of linear equations by transforming the system's augmented matrix into row echelon form.

Back Substitution: A process used to solve an upper triangular system of equations, starting from the last equation and working upwards.

Row Echelon Form: A form of a matrix where each leading entry (the first non-zero number from the left) in a row is to the right of any leading entry in the previous row.