Question

If $A = \left[ {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  2&3&0&0 \\
  4&5&6&0 \\
  7&8&9&{10}
\end{array}} \right]$, then $A$ is
A. An upper triangular matrix
B. A null matrix
C. A lower triangular matrix
D. None of these

Answer 1

Hint: In this question, we have to check the type of the given matrix. 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 check the condition of each matrix given in the options with the given matrix $A$ to find the correct solution to the given problem.

Complete step by step Solution:
Given matrix is $A = \left[ {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  2&3&0&0 \\
  4&5&6&0 \\
  7&8&9&{10}
\end{array}} \right]$
Elements of matrix $A$ are
\[
  {a_{11}} = 1,{a_{12}} = 0,{a_{13}} = 0,{a_{14}} = 0, \\
  {a_{21}} = 2,{a_{22}} = 3,{a_{23}} = 0,{a_{24}} = 0, \\
  {a_{31}} = 4,{a_{32}} = 5,{a_{33}} = 6,{a_{34}} = 0, \\
  {a_{41}} = 7,{a_{42}} = 8,{a_{43}} = 9,{a_{44}} = 10 \\
 \]

Checking the condition for option A:
If \[{a_{ij}} = 0;i > j,\] then the matrix is known as “an upper triangular matrix”.
The condition for being an upper triangular matrix is not satisfying as \[{a_{21}} \ne 0,{a_{31}} \ne 0,{a_{32}} \ne 0,{a_{41}} \ne 0,{a_{42}} \ne 0,{a_{43}} \ne 0\]
\[(\because \] For \[{a_{21}},2 > 1\], for \[{a_{31}},3 > 1\],for \[{a_{32}},3 > 2\] , for \[{a_{41}},4 > 1\], for \[{a_{42}},4 > 2\], for \[{a_{43}},4 > 3)\]
So, matrix $A$ is not an upper triangular matrix.

Checking the condition for option B:
 If \[{a_{ij}} = 0,\] then the matrix is known as “a null matrix”.
The condition for being a null matrix is not satisfying as all the elements of the matrix $A$ are not zero:
\[{a_{11}} \ne 0,{a_{21}} \ne 0,{a_{22}} \ne 0,{a_{31}} \ne 0,{a_{32}} \ne 0,{a_{33}} \ne 0,{a_{41}} \ne 0,{a_{42}} \ne 0,{a_{43}} \ne 0,{a_{44}} \ne 0\]
So, matrix $A$ is not a null matrix.

Checking the condition for option C:
If \[{a_{ij}} = 0;i < j,\] then the matrix is known as “a lower triangular matrix”.
The condition for being a lower triangular matrix is completely satisfying as \[{a_{12}} = 0,{a_{13}} = 0,{a_{14}} = 0,{a_{23}} = 0,{a_{24}} = 0,{a_{34}} = 0\]
\[(\because \] For \[{a_{12}},1 < 2\], for \[{a_{13}},1 < 3\], for \[{a_{14}},1 < 4\], for \[{a_{23}},2 < 3\], for \[{a_{24}},2 < 4\], for \[{a_{34}},3 < 4)\]
So, matrix $A$ is a lower triangular matrix.

Therefore, the correct option is (C).

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 definition of each type of matrices, it is necessary to match the given matrix with the conditions of all the types of matrices given in the options to get the correct answer. Conditions must be checked very carefully.