Question

Square matrix ${\left[ {{a_{ij}}} \right]_{n \times n}}$, will be an upper triangular matrix if
A. ${a_{ij}} \ne 0,$for $i > j$
B. ${a_{ij}} = 0,$for $i > j$
C. ${a_{ij}} = 0,$for $i < j$
D. None of these

Answer 1

Hint: In this question, we have to find out whether the given matrix is an upper triangular matrix or not. To find the answer to the given question, types of matrices should be known, such as row matrix, column matrix, null matrix, square matrix, upper triangular matrix, lower triangular matrix, etc. In this question, we will match the conditions given in the options with the condition of an upper triangular matrix to find the correct solution to the given problem.

Complete step by step Solution:
A square matrix is said to be an upper triangular matrix when all of its elements below the principal diagonal are zero.
For example, consider the below-given $3 \times 3$ matrix:
$A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\
  0&4&5 \\
  0&0&6
\end{array}} \right]$
Elements of matrix $A$are:
\[
  {a_{11}} = 1,{a_{12}} = 2,{a_{13}} = 3, \\
  {a_{21}} = 0,{a_{22}} = 4,{a_{23}} = 5, \\
  {a_{31}} = 0,{a_{32}} = 0,{a_{33}} = 6 \\
 \]
Principal diagonal elements are:
\[{a_{11}} = 1,{a_{22}} = 4,{a_{33}} = 6\]
The elements below these principal diagonal elements are
\[{a_{21}} = 0,{a_{31}} = 0,{a_{32}} = 0\]
It is clearly visible that all the elements below the principal diagonal are zero. Also, for \[{a_{21}},2 > 1,\]for\[{a_{31}},3 > 1,\]and for \[{a_{32}},3 > 2\], hence, it shows that the condition for being an upper triangular matrix should be \[{a_{ij}} = 0\], where \[i > j\].

Therefore, the correct option is (B).

Additional Information: In this type of question, where we have to focus on the conditions of the matrices, we must have a clear knowledge of the indices of the matrix and its representation, i.e., \[{a_{ij}},\] where \[i\] represents the row and \[j\] represents the column which we are referring to.

Note: Since the problem is based on types of matrices, hence, even after knowing the condition of each type of matrices, it is necessary to take an example as per the definition of the matrix mentioned in the question to get the correct answer. Be careful while checking the ‘greater than’ and ‘smaller than symbols \[( < and > )\].