
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
164.4k+ views
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.
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.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Instantaneous Velocity - Formula based Examples for JEE

JEE Advanced 2025 Notes

JEE Main Chemistry Question Paper with Answer Keys and Solutions

Degree of Dissociation and Its Formula With Solved Example for JEE
