Upper triangular matrix is a square matrix whose lower off-diagonal elements are zero. It is usually denoted by the capital letter ‘U‘.
A square matrix P = [xij] is said to be upper triangular matrix (UTM) if xij = 0 when i > j.
Note: In such matrix, the diagonal and/or upper off-diagonal elements may or may not be zero.
Upper triangular matrix examples
The example of UTM of order 3 (or matrix size is 3 x 3) is given below.
![\[U=\begin{bmatrix} 1 &2 &3 \\ 0& 4 &5 \\ 0&0 & 6 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-25725325744d3ff52b62295864da9326_l3.png "Rendered by QuickLaTeX.com")
The example of UTM of order 2 (or matrix size is 2 x 2) is given below.
![\[U=\begin{bmatrix} 1 &3 \\ 0& 1 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-fcffd0eb524217355e6b8b62374fede4_l3.png "Rendered by QuickLaTeX.com")
Upper triangular matrix determinant
The determinant of an upper triangular matrix is the product of its diagonal elements.
Let us understand by taking an example. Suppose U is an upper triangular matrix given as
![\[U=\begin{bmatrix} 7 &4 &3 \\ 0& 6 &1 \\ 0&0 & 3 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-da1e1af695e3fbec762e6deccc658c1b_l3.png "Rendered by QuickLaTeX.com")
Then the determinant of this matrix U is calculated as the product of the diagonal elements. Here, in this case, the diagonal elements are 7, 6 and 3. So the determinant is equal to 7 × 6 × 3 = 126.
i.e. |U| = 7 × 6 × 3 = 126
