Question

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then Adj(Adj(Adj A)) =

• A. A
• B. $A^{-1}$
• C. $|A|A^{-1}$
• D. $\frac{A^{-1}}{|A|}$

Hint:

Memorizing the key properties of the adjoint matrix is crucial. For a matrix of order $n$, $\text{Adj}(\text{Adj A}) = |A|^{n-2}A$. This formula simplifies significantly for $n=2$ to just $A$, and for $n=3$ to $|A|A$. Being able to quickly apply these saves a lot of time compared to computing the adjoints manually.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
This question involves properties of the adjoint of a matrix. The adjoint of a matrix \( A \), denoted as \( \text{Adj}(A) \), is the transpose of its cofactor matrix. We will use standard properties relating the adjoint, determinant, and inverse of a matrix.

Step 2: Key Formula or Approach
We will use the following properties for a non-singular square matrix \( A \) of order \( n \):
1. \( A(\text{Adj} A) = (\text{Adj} A)A = |A|I \)
2. \( A^{-1} = \frac{1}{|A|} \text{Adj} A \implies \text{Adj} A = |A|A^{-1} \)
3. \( \text{Adj(Adj A)} = |A|^{n-2} A \)

Step 3: Detailed Explanation
The given matrix is \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), which is a square matrix of order \( n = 2 \).
We need to find \( \text{Adj(Adj(Adj A))} \).

First, find \( \text{Adj(Adj A)} \) using property (3):
\[ \text{Adj(Adj A)} = |A|^{2-2} A = |A|^0 A = A \] Hence, the expression simplifies to:
\[ \text{Adj(Adj(Adj A))} = \text{Adj}(A) \] Now, we need to find \( \text{Adj}(A) \).
For a \( 2 \times 2 \) matrix \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), we have \( \text{Adj}(M) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \).
So, for our matrix \( A \):
\[ \text{Adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \] Therefore, \( \text{Adj(Adj(Adj A))} = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \).

Now, let's verify using property (2): \( \text{Adj}(A) = |A|A^{-1} \).
Calculate the determinant of \( A \):
\[ |A| = (1)(4) - (2)(3) = 4 - 6 = -2 \] Since \( |A| \neq 0 \), \( A^{-1} \) exists.
Using property (2):
\[ \text{Adj}(A) = |A|A^{-1} \] Since we found \( \text{Adj(Adj(Adj A))} = \text{Adj}(A) \), we can conclude:
\[ \text{Adj(Adj(Adj A))} = |A|A^{-1} \] This matches option (C).

Step 4: Final Answer
Using the properties of adjoint matrices for a matrix of order \( n = 2 \), we have \( \text{Adj(Adj A)} = A \). Therefore, \( \text{Adj(Adj(Adj A))} = \text{Adj}(A) \). Also, \( \text{Adj}(A) = |A|A^{-1} \). Hence, the correct expression is: \[ \boxed{\text{Adj(Adj(Adj A))} = |A|A^{-1}} \]