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

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: -

& \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{aligned}$$ Taking determinant both sides, we get, $$\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{aligned} & \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{aligned}$$ Now, substituting the value of $$\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{aligned} & \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{aligned}$$ **Hence, option (d) is the correct answer.** **Note:** One must remember the properties of the determinant of a matrix and its adjoint. The formula, $$\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 $$.