Question

Let \(A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1\\ 1 & 3 & 5 \end{bmatrix}\). Then the sum of all elements of the matrix \(\operatorname{adj}(\operatorname{adj}(2(\operatorname{adj}A)^{-1}))\) is equal to:

• A. 3
• B. 4
• C. -4
• D. -3

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to evaluate the given matrix expression using the properties of adjoints and inverses.
The key properties are:
1. \(A^{-1} = \frac{\text{adj}(A)}{|A|} \implies (\text{adj} A)^{-1} = \frac{A}{|A|}\)
2. \(\text{adj}(\text{adj} X) = |X|^{n-2} X\) for an \(n \times n\) matrix \(X\).
Step 2: Key Formula or Approach:
Let's first calculate the determinant of \(A\):
\[ |A| = 1(0 - 3) - 1(-10 - 1) + 2(-6 - 0) = -3 + 11 - 12 = -4 \] Now, simplify the innermost expression \(B = 2(\text{adj} A)^{-1}\):
\[ B = 2 \left( \frac{A}{|A|} \right) = \frac{2}{-4} A = -\frac{1}{2} A \] Step 3: Detailed Explanation:
We now need to find \(\text{adj}(\text{adj} B)\) for the \(3 \times 3\) matrix \(B\).
Using the property \(\text{adj}(\text{adj} B) = |B|^{3-2} B = |B| B\).
Let's find the determinant of \(B\):
\[ |B| = \left| -\frac{1}{2} A \right| = \left(-\frac{1}{2}\right)^3 |A| = -\frac{1}{8} \times (-4) = \frac{1}{2} \] So, the required matrix is:
\[ |B| B = \frac{1}{2} \left(-\frac{1}{2} A\right) = -\frac{1}{4} A \] Now, we calculate the sum of all elements of \(A\):
Sum = \((1 + 1 + 2) + (-2 + 0 + 1) + (1 + 3 + 5) = 4 - 1 + 9 = 12\).
Step 4: Final Answer:
The required matrix is \(-\frac{1}{4} A\). The sum of all its elements is:
\[ -\frac{1}{4} \times (\text{Sum of elements of } A) = -\frac{1}{4} \times 12 = -3 \]