SATHEE: Matrices and Determinants 1 Question 12

Matrices and Determinants 1 Question 12

12. For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is/are not correct?

(2013 Adv.)

(a) $N^{T} M N$ is symmetric or skew-symmetric, according as $M$ is symmetric or skew-symmetric

(b) $M N - N M$ is symmetric for all symmetric matrices $M$ and $N$

(d) $(adj M) (adj N) = adj (M N)$ for all invertible matrices $M$ and $N$

Show Answer

Answer:

Correct Answer: 12. (c,d)

Solution:

(a) ${(N^{T} M N)}^{T} = N^{T} M^{T} {(N^{T})}^{T} = N^{T} M^{T} N$ , is symmetric if $M$ is symmetric and skew-symmetric, if $M$ is skew-symmetric.

$(b) \begin{aligned} (M N - N M)^{T} & = (M N)^{T} - (N M)^{T} & = N M - M N = - (M N - N M) \end{aligned}$

$∴$ Skew-symmetric, when $M$ and $N$ are symmetric.

$∴$ Not correct.

(d) $(adj M N) = (adj N) \cdot (adj M)$

$∴$ Not correct.

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Matrices and Determinants 1 Question 12

12. For 3×3 matrices M and N, which of the following statement(s) is/are not correct?

Answer:

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12. For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is/are not correct?