Question

Which among the following are triangular matrices?

• A. $\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}5& 0& 0\\ 1& 2& 0\\ 0& 3& 3\end{array}\right]$
• C. $\left[\begin{array}{ccc}1& 3& 0\\ 2& 1& 3\\ 0& 2& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}2& 3& 9\\ 1& 5& 0\\ 6& 0& 0\end{array}\right]$

Answer 1

Answer: (A), (B)

The correct options are
A

$\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$

B

$\left[\begin{array}{ccc}5& 0& 0\\ 1& 2& 0\\ 0& 3& 3\end{array}\right]$

Try to remember the definitions of upper and lower triangular matrices. A matrix which is either upper or lower triangular are called in general as triangular matrix.

Also note that in definition we say elements above or below the main diagonals are to be zeroes for lower and upper triangular matrices respectively. We do not say the other elements are to be necessarily non zero. Option (b) satisfies the definition of lower triangular matrix although $a_{31}=0$ .

Similarly definition is based on the definition of main diagonal (which is the set of all elements aij where i=j). Not any other diagonal as in the case of option (d).

Hence you can see option (a) and (b) are clearly triangular matrices.