What is Upper Triangular Matrix? Determinant & Examples

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.

    \[U=\begin{bmatrix} 1 &2 &3 \\ 0& 4 &5 \\ 0&0 & 6 \end{bmatrix}\]

The example of UTM of order 2 (or matrix size is 2 x 2) is given below.

    \[U=\begin{bmatrix} 1 &3 \\ 0& 1 \end{bmatrix}\]

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

    \[U=\begin{bmatrix} 7 &4 &3 \\ 0& 6 &1 \\ 0&0 & 3 \end{bmatrix}\]

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

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