If \(A = [a_{ij}]_{2\times 2}\) where \(a_{ij} = \begin{cases}1, & i \ne j\\ 0, & i = j\end{cases}\), and \(I\) is the identity matrix of order 2, then \((A^2-3A +4I)\) is
(A) Symmetric Matrix
(B) Skew-symmetric Matrix
(C) Non-singular Matrix
(D) Square Matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
The diagonal entries are 0 and the other entries are 1:
\( \displaystyle A = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix} \)
\( \displaystyle A^2 = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix} = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix} = I \)
Let \(B = A^2 - 3A + 4I\).
\(B = I - 3A + 4I = 5I - 3A\)
\( \displaystyle B = \begin{bmatrix}5 & -3\\ -3 & 5\end{bmatrix} \)
So B is the 2 × 2 matrix with diagonal entries 5, 5 and off-diagonal entries −3, −3.
Checking the statements
(A) Symmetric Matrix — Correct. The off-diagonal entries are equal (−3 and −3), so \(B^T = B\).
(B) Skew-symmetric Matrix — Incorrect. A skew-symmetric matrix has zero diagonal, but the diagonal of B is 5, 5.
(C) Non-singular Matrix — Correct. \(|B| = 5 \times 5 - (-3)(-3) = 25 - 9 = 16 \ne 0\).
(D) Square Matrix — Correct. B is of order 2 × 2.