---

Solution

List I		List II
A.	Kailash Satyarthi	II.	Peace
B.	Abhijit Banerjee	IV.	Economics
C.	Venkatraman Ramakrishnan	I.	Chemistry
D.	Subrahmanyan Chandrasekhar	III.	Physics

---

Solution

---

Solution

---

Solution

Solution (Match the Columns):

(A) \(\begin{vmatrix}\lambda-1 & 0 \\ 0 & \lambda-1\end{vmatrix}=(\lambda-1)^2\). Singular ⇒ \((\lambda-1)^2=0 \Rightarrow \lambda=1\). ⇒ (A → II)

(B) \(\Delta=\begin{vmatrix}1 & 2 \\ 2 & 4\end{vmatrix}=1\cdot4-2\cdot2=0\). ⇒ (B → I)

(C) \(A=\begin{bmatrix}1&0\\0&\tfrac12\end{bmatrix}\), so \(|A|=\tfrac12\). ⇒ \(|A^{-1}|=1/|A|=2\). ⇒ (C → IV)

(D) \(\begin{bmatrix}a+1&1\\1&2\end{bmatrix}=\begin{bmatrix}-1&1\\1&2\end{bmatrix}\). ⇒ \(a+1=-1 \Rightarrow a=-2\). ⇒ (D → III)

✅ Correct option: (1) — (A-II), (B-I), (C-IV), (D-III)

---

Solution

---

Solution

---

Solution

---

Solution

A(BA) = (AB)A = ABA

If A and B are symmetric matrices, then

(ABA)^T = A^TB^TA^T

Since A and B are symmetric,

A^T = A and B^T = B

Therefore

(ABA)^T = ABA

Hence A(BA) and (AB)A are symmetric matrices.

Also, AB is symmetric if AB = BA (i.e., A and B commute).

Correct Answer: (a) Assertion is true, Reason is true, and Reason is the correct explanation of the Assertion.

---

Solution

If a matrix A is symmetric, then
A^T = A

If a matrix A is skew symmetric, then
A^T = −A

For both conditions to hold:
A = −A

⇒ 2A = 0
⇒ A = 0

Therefore, the matrix must be a zero matrix.

Correct Answer: (b) A is zero matrix

---

Solution