NDA 1 2025 Maths, question 27

Topic: Matrices and determinants

Q27

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ , then what is $A^{2} - 4A$ equal to? where $I_{3}$ is the identity matrix of order 3.

(a) $- 5I_{3}$
(b) $- I_{3}$
(c) $I_{3}$
(d) $5I_{3}$ Correct answer

Solution.

Answer: (d) $5I_{3}$

Each diagonal entry of $A^{2}$ is $1^{2} + 2^{2} + 2^{2} = 9$ .
Each off-diagonal entry is $1 \times 2 + 2 \times 1 + 2 \times 2 = 8$ .
Subtracting $4A$ leaves diagonal entries $9 - 4 = 5$ and off-diagonal entries $8 - 8 = 0$ .
Hence $A^{2} - 4A = 5I_{3}$ .