Answer
Verified
407.4k+ views
- Hint: Check if the matrix is not nilpotent by finding the determinant of the given matrix. If the determinant is non-zero, then the matrix is not nilpotent. If the determinant is zero, find ${{A}^{2}},{{A}^{3}},\cdots $ successively and hence find the smallest value of k such that ${{A}^{k}}=O$ and hence find the order of the nilpotent matrix.
Complete step-by-step solution -
Nilpotent matrix: A square matrix A is said to be a nilpotent matrix if there exists $k\in \mathbb{N}$, such that ${{A}^{k}}=O$ where O is a null matrix of the same dimensions as of A. The smallest value of k such that ${{A}^{k}}=O$ is called the order of the nilpotent matrix. Hence if k is the order of the nilpotent matrix, then ${{A}^{n}}=O\forall n\ge k$ and ${{A}^{n}}\ne O\forall n$\det \left( {{A}^{k}} \right)=0$
Since $\det \left( {{A}^{k}} \right)={{\left( \det A \right)}^{k}}$, we get
$\det \left( A \right)=0$
Hence a matrix can be nilpotent if $\det \left( A \right)=0$ and if $\det \left( A \right)\ne 0$ then the matrix cannot be nilpotent.
In the given matrix, we have
$A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$
Hence $\det \left( A \right)=1\left( -6+6 \right)-1\left( -15+12 \right)+3\left( -5+4 \right)=0+3-3=0$
Since det(A) = 0, the matrix can be a nilpotent matrix.
Now, we have
$A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$
Hence ${{A}^{1}}\ne O$.
${{A}^{2}}=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]=\left[ \begin{matrix}
0 & 0 & 0 \\
3 & 3 & 9 \\
-1 & -1 & -3 \\
\end{matrix} \right]$
Since ${{A}^{2}}\ne O$ we have $k\ne 0$
Now, we have ${{A}^{3}}={{A}^{2}}A$
Hence, we have
${{A}^{3}}=\left[ \begin{matrix}
0 & 0 & 0 \\
3 & 3 & 9 \\
-1 & -1 & -3 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]=\left[ \begin{matrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{matrix} \right]=O$
Since ${{A}^{3}}=O$, we have the order of nilpotency = 3.
Hence the order of the given nilpotent matrix is 3, and hence option [b] is correct.
Note: [1] A nilpotent matrix has only 0 as its eigenvalue, and hence characteristic polynomial equation is ${{x}^{n}}=0$. The two statements are equivalent, i.e. A matrix is nilpotent if ${{x}^{n}}=0$ is the characteristic polynomial equation of the matrix and if the matrix is nilpotent, then ${{x}^{n}}=0$ is the characteristic polynomial equation.
The characteristic polynomial equation of a matrix A is the polynomial equation $\det \left( A-xI \right)=0$.
Hence we can prove that $A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$ is a nilpotent matrix by verifying $\det \left( A-xI \right)=0$ is the polynomial equation ${{x}^{3}}=0$
Complete step-by-step solution -
Nilpotent matrix: A square matrix A is said to be a nilpotent matrix if there exists $k\in \mathbb{N}$, such that ${{A}^{k}}=O$ where O is a null matrix of the same dimensions as of A. The smallest value of k such that ${{A}^{k}}=O$ is called the order of the nilpotent matrix. Hence if k is the order of the nilpotent matrix, then ${{A}^{n}}=O\forall n\ge k$ and ${{A}^{n}}\ne O\forall n
Since $\det \left( {{A}^{k}} \right)={{\left( \det A \right)}^{k}}$, we get
$\det \left( A \right)=0$
Hence a matrix can be nilpotent if $\det \left( A \right)=0$ and if $\det \left( A \right)\ne 0$ then the matrix cannot be nilpotent.
In the given matrix, we have
$A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$
Hence $\det \left( A \right)=1\left( -6+6 \right)-1\left( -15+12 \right)+3\left( -5+4 \right)=0+3-3=0$
Since det(A) = 0, the matrix can be a nilpotent matrix.
Now, we have
$A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$
Hence ${{A}^{1}}\ne O$.
${{A}^{2}}=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]=\left[ \begin{matrix}
0 & 0 & 0 \\
3 & 3 & 9 \\
-1 & -1 & -3 \\
\end{matrix} \right]$
Since ${{A}^{2}}\ne O$ we have $k\ne 0$
Now, we have ${{A}^{3}}={{A}^{2}}A$
Hence, we have
${{A}^{3}}=\left[ \begin{matrix}
0 & 0 & 0 \\
3 & 3 & 9 \\
-1 & -1 & -3 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]=\left[ \begin{matrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{matrix} \right]=O$
Since ${{A}^{3}}=O$, we have the order of nilpotency = 3.
Hence the order of the given nilpotent matrix is 3, and hence option [b] is correct.
Note: [1] A nilpotent matrix has only 0 as its eigenvalue, and hence characteristic polynomial equation is ${{x}^{n}}=0$. The two statements are equivalent, i.e. A matrix is nilpotent if ${{x}^{n}}=0$ is the characteristic polynomial equation of the matrix and if the matrix is nilpotent, then ${{x}^{n}}=0$ is the characteristic polynomial equation.
The characteristic polynomial equation of a matrix A is the polynomial equation $\det \left( A-xI \right)=0$.
Hence we can prove that $A=\left[ \begin{matrix}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3 \\
\end{matrix} \right]$ is a nilpotent matrix by verifying $\det \left( A-xI \right)=0$ is the polynomial equation ${{x}^{3}}=0$
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE