Upper triangular form

Definition

A matrix is in upper triangular form if all the entries below the main diagonal are zero. This form simplifies solving systems of linear equations using back-substitution.

5 Must Know Facts For Your Next Test

1. An upper triangular matrix has non-zero elements only on or above the main diagonal.
2. Upper triangular form is often achieved through Gaussian elimination.
3. Solving a system of equations with an upper triangular matrix involves back-substitution.
4. The determinant of an upper triangular matrix is the product of its diagonal entries.
5. If a square matrix can be transformed into an upper triangular form without row exchanges, it indicates that the system has a unique solution.

Review Questions

• What characteristics define an upper triangular matrix?
• How does transforming a system into upper triangular form help in solving it?
• Explain the process and purpose of back-substitution in solving systems with upper triangular matrices.

Related terms

Gaussian Elimination: A method to solve systems of linear equations by transforming the coefficient matrix into an upper triangular form.

Back-Substitution: A technique used to solve a system of linear equations after it has been transformed into an upper triangular form by starting from the last equation and substituting backwards.

Main Diagonal: The diagonal running from the top-left to bottom-right of a square matrix, consisting of elements where the row and column indices are equal.