Question

Define (1) An upper triangular matrix. (2) A lower triangular matrix. (3) A properly triangular matrix. Give examples.

Short answer

An upper triangular matrix is a square matrix with all elements below the main diagonal being zero, e.g., \[ \begin{bmatrix} 4 & 3 & 2 \\ 0 & 1 & 5 \\ 0 & 0 & 6 \end{bmatrix} \] A lower triangular matrix is a square matrix with all elements above the main diagonal being zero, e.g., \[ \begin{bmatrix} 7 & 0 & 0 \\ 8 & 3 & 0 \\ 1 & 6 & 9 \end{bmatrix} \] A properly triangular matrix is a triangular matrix where all main diagonal elements are also zero, e.g., upper properly triangular matrix: \[ \begin{bmatrix} 0 & 5 & 4 \\ 0 & 0 & 8 \\ 0 & 0 & 0 \end{bmatrix} \] and lower properly triangular matrix: \[ \begin{bmatrix} 0 & 0 & 0 \\ 4 & 0 & 0 \\ 9 & 2 & 0 \end{bmatrix} \]

Step-by-step solution

Define Upper Triangular Matrix

An upper triangular matrix is a square matrix where all the elements below the main diagonal are zero. The main diagonal consists of elements with equal row and column indices, and lies from the top-left to the bottom-right corner of the matrix.

Define Lower Triangular Matrix

A lower triangular matrix is a square matrix where all the elements above the main diagonal are zero. Similar to the upper triangular matrix, the main diagonal consists of elements with equal row and column indices, running diagonally from the top-left to the bottom-right corner of the matrix.

Define Properly Triangular Matrix

A properly triangular matrix is a type of triangular matrix in which all the elements on the main diagonal are also zero. Essentially, a properly triangular matrix can be either a lower triangular matrix or an upper triangular matrix with all diagonal elements being zero.

Provide Examples

Now, let's provide examples for each type of triangular matrix. Upper Triangular Matrix Example: \[ \begin{bmatrix} 4 & 3 & 2 \\ 0 & 1 & 5 \\ 0 & 0 & 6 \end{bmatrix} \] Lower Triangular Matrix Example: \[ \begin{bmatrix} 7 & 0 & 0 \\ 8 & 3 & 0 \\ 1 & 6 & 9 \end{bmatrix} \] Properly Triangular Matrix Example (Upper Properly Triangular Matrix): \[ \begin{bmatrix} 0 & 5 & 4 \\ 0 & 0 & 8 \\ 0 & 0 & 0 \end{bmatrix} \] Properly Triangular Matrix Example (Lower Properly Triangular Matrix): \[ \begin{bmatrix} 0 & 0 & 0 \\ 4 & 0 & 0 \\ 9 & 2 & 0 \end{bmatrix} \]