| List I | List II | ||
|---|---|---|---|
| A. | Kailash Satyarthi | I. | Chemistry |
| B. | Abhijit Banerjee | II. | Peace |
| C. | Vinkatraman Ramakrishnan | III. | Physics |
| D. | Subrahmanyan Chandrasekhar | IV. | Economics |
| List I | List II | ||
|---|---|---|---|
| A. | Kailash Satyarthi | II. | Peace |
| B. | Abhijit Banerjee | IV. | Economics |
| C. | Venkatraman Ramakrishnan | I. | Chemistry |
| D. | Subrahmanyan Chandrasekhar | III. | Physics |
| List-I | List-II |
| (A) If $\begin{vmatrix}\lambda-1&0\\ 0&\lambda-1\end{vmatrix}$ then $\lambda=$ | (I) 0 |
| (B) If $\Delta=\begin{vmatrix}1&2\\ 2&4\end{vmatrix}$ then $\Delta$ is | (II) 1 |
| (C) If $A=\begin{bmatrix}1&0\\ 0&\frac{1}{2}\end{bmatrix}$ then $|A^{-1}|$ is | (III) -2 |
| (D) If $\begin{bmatrix}a+1&1\\ 1&2\end{bmatrix}=\begin{bmatrix}-1&1\\ 1&2\end{bmatrix}$ then a is | (IV) 2 |
Assertion: A(BA) and (AB)A are symmetric matrices.
Reason: AB is symmetric matrix, if matrix multiplication of A with B is commutation.
Options:
(a) Assertion is true, Reason is true, and Reason is the correct explanation of the Assertion.
(b) Assertion is true, Reason is true, but Reason is not the correct explanation of the Assertion.
(c) Assertion is true, but Reason is false.
(d) Assertion is false, but Reason is true.
A(BA) = (AB)A = ABA
If A and B are symmetric matrices, then
(ABA)T = ATBTAT
Since A and B are symmetric,
AT = A and BT = 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.
(a) A is diagonal matrix
(b) A is zero matrix
(c) A is scalar matrix
(d) A is square matrix
If a matrix A is symmetric, then
AT = A
If a matrix A is skew symmetric, then
AT = −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
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Syllabus, Notification
and More.