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: Key Property of Adjoint: For a non-singular square matrix \( A \) of order \( n \): \[ \text{adj}(\text{adj}(A)) = |A|^{n-2} A \]
Step 2: Applying to the Problem: Here, \( A \) is a \( 2 \times 2 \) matrix, so \( n=2 \). \[ \text{adj}(\text{adj}(A)) = |A|^{2-2} A = |A|^0 A = A \] Now we need to find \( \text{Adj}(\text{Adj}(\text{Adj } A)) \). Let \( B = \text{Adj}(A) \). Then we need \( \text{Adj}(\text{Adj}(B)) \). Using the same property for matrix \( B \): \[ \text{Adj}(\text{Adj}(B)) = B \] Substituting back \( B = \text{Adj}(A) \): \[ \text{Adj}(\text{Adj}(\text{Adj } A)) = \text{Adj}(A) \]
Step 3: Relate Adj(A) to options: We know that \( A^{-1} = \frac{1}{|A|} \text{Adj}(A) \). Multiplying by \( |A| \): \[ \text{Adj}(A) = |A| A^{-1} \] Thus, the expression simplifies to \( |A| A^{-1} \).