22.5.4. Upper or Lower Triangular Matrix

Try these first

1. 1.

   Identify the following matrix as upper or lower triangular: $\begin{pmatrix} 1 & 3 & 5 \\ 0 & 2 & 4 \\ 0 & 0 & 0 \end{pmatrix}$

   Hint

   Look where the non-zero elements are located.

2. 2.

   What is the rank of the following matrix? $\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 0 \\ 0 & 3 & 1 \end{pmatrix}$

   Hint

   Count the number of non-zero rows.

3. 3.

   What defines an upper triangular matrix?

   • A matrix with all zeroes above the diagonal.
   • A matrix with all zeroes below the diagonal.
   • A matrix with non-zero elements in the first column.

   Hint

   Think about where the non-zero elements are in the matrix.

4. 4.

   The rank of a triangular matrix is equal to the number of non-zero rows.

   • True
   • False

   Hint

   Recall the definition of rank in the context of triangular matrices.

5. 5.

   Analyze the following upper triangular matrix: $\begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{pmatrix}$. What is its rank and why?

   Hint

   Count the rows that contain non-zero elements.

6. 6.

   Construct a lower triangular matrix of order 4 with a rank of 3 and show your work.

   Hint

   Ensure there are three non-zero rows in your matrix.