What is Lower Triangular Matrix? Determinant & Examples

Lower triangular matrix is a square matrix whose upper off-diagonal elements are zero. It is usually denoted by the capital letter ‘L‘.

A square matrix Q = [xij] is said to be lower triangular matrix (LTM) if xij = 0 when i < j.

Note: In this matrix, the diagonal and/or lower off-diagonal elements may or may not be zero.

Lower triangular matrix examples

The example of LTM of order 3 (or matrix size is 3 x 3) is given below.

    \[L=\begin{bmatrix} 6 &0 &0 \\ -4& 8 &0 \\ 2&3 & 6 \end{bmatrix}\]

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

    \[L=\begin{bmatrix} 1 &0 \\ 4& 2 \end{bmatrix}\]

Lower triangular matrix determinant

The determinant of a lower triangular matrix is the product of its diagonal elements.

Let us understand by taking an example. Suppose L is a lower triangular matrix given as

    \[L=\begin{bmatrix} 4 &0 &0 \\ -4& 3 &0 \\ 2&3 & 2 \end{bmatrix}\]

Then the determinant of this matrix L is calculated as the product of the diagonal elements. Here, in this case, the diagonal elements are 4, 3 and 2. So the determinant is equal to 4 × 3 × 2 = 24.

i.e. |L| = 4 × 3 × 2 = 24

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