Question

The characteristic equation of a matrix A is \[{{\Omega }^{3}}-5{{\Omega }^{2}}-3\Omega +2=0\], then \[\left| adj\left( A \right) \right|=\]
(a) 9
(b) 25
(c) \[\dfrac{1}{2}\]
(d) 4

Answer 1

Hint: Equate the given characteristic equation of the matrix A with the expression det. \[\left[ A-\Omega I \right]\], which is the determinant of the matrix \[\left( A-\Omega I \right)\]. Here, \[\Omega \] is called the eigen value and I is called the identity matrix. Equate \[\left| A-\Omega I \right|={{\Omega }^{3}}-5{{\Omega }^{2}}-3\Omega +2\] and substitute \[\Omega =0\] to find the value of determinant of matrix A, |A|. Now, use the relation between determinant of A and determinant of adjoint of A given as \[\left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}}\] to get the answer. Here, ‘n’ is the order of the matrix. To determine the value of ‘n’ check the highest power of \[\Omega \].

Complete step-by-step solution
Here, we have been provided with the characteristic equation of a matrix A given as: - \[{{\Omega }^{3}}-5{{\Omega }^{2}}-3\Omega +2=0\]. Here, \[\Omega \] is called the eigenvalue.
Now, we know that the characteristic equation of a matrix A is given as the determinant of the matrix \[\left( A-\Omega I \right)\], where “I” is the identity matrix. So, mathematically, we have,
\[\Rightarrow \] Characteristic equation of matrix A = \[\left| A-\Omega I \right|\]
Equating it with the given characteristic equation, we get,
\[\Rightarrow \left| A-\Omega I \right|={{\Omega }^{3}}-5{{\Omega }^{2}}-3\Omega +2\]
Substituting \[\Omega =0\], we get,
\[\Rightarrow \left| A-0.I \right|=2\]
\[\Rightarrow \left| A \right|=2\] - (1)
Now, we have to find the value of expression \[\left| adj\left( A \right) \right|\], which is the determinant of adjoint of matrix A. So, let us derive an expression for \[\left| adj\left( A \right) \right|\].
We know that inverse of a matrix A is given as: -
\[\begin{align}
  & \Rightarrow {{A}^{-1}}=\dfrac{adj\left( A \right)}{\left| A \right|} \\
 & \Rightarrow A.{{A}^{-1}}=\dfrac{A.adj\left( A \right)}{\left| A \right|} \\
 & \Rightarrow I=\dfrac{A.adj\left( A \right)}{\left| A \right|} \\
 & \Rightarrow A.adj\left( A \right)=\left| A \right|.I \\
\end{align}\]
Taking determinant both sides, we get,
\[\Rightarrow \left| A.adj\left( A \right) \right|=\left| \left| A \right|.I \right|\]
Now, if A is a matrix of order n then we have,
\[\begin{align}
  & \Rightarrow \left| A.adj\left( A \right) \right|={{\left| A \right|}^{n}} \\
 & \Rightarrow \left| A \right|.\left| adj\left( A \right) \right|={{\left| A \right|}^{n}} \\
 & \Rightarrow \left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}} \\
\end{align}\]
Now, substituting the value of \[\left| A \right|\] from equation (1) in the above relation, we get,
\[\Rightarrow \left| adj\left( A \right) \right|={{2}^{n-1}}\]
In the characteristic equation, we can clearly see that the highest power of \[\Omega \] is 3. This is only possible if the order of the given matrix is 3. So, n = 3.
\[\begin{align}
  & \Rightarrow \left| adj\left( A \right) \right|={{2}^{3-1}} \\
 & \Rightarrow \left| adj\left( A \right) \right|={{2}^{2}} \\
 & \Rightarrow \left| adj\left( A \right) \right|=4 \\
\end{align}\]
Hence, option (d) is the correct answer.

Note: One must remember the properties of the determinant of a matrix and its adjoint. The formula, \[\left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}}\] is used directly in many places without derivation. So, it must be remembered. Note that whatever operations we are performing is taking place on a square matrix of order 3. This is because the determinant of a non – square matrix is meaningless. Remember that the order of the matrix will be the highest power of eigenvalue \[\Omega \].