SATHEE: Matrices and Determinants 2 Question 22

Matrices and Determinants 2 Question 22

24. Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M$ . Further, if $M \neq N^{2}$ and $M^{2} = N^{4}$ , then

(2014 Adv.)

(a) determinant of $(M^{2} + M N^{2})$ is 0

(b) there is a $3 \times 3$ non-zero matrix $U$ such that $(M^{2} + M N^{2}) U$ is zero matrix

(d) for a $3 \times 3$ matrix $U$ , if $(M^{2} + M N^{2}) U$ equals the zero matrix, then $U$ is the zero matrix

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Answer:

Correct Answer: 24. (a, b)

Solution:

PLAN: (i) If $A$ and $B$ are two non-zero matrices and $A B = B A$ , then $(A - B) (A + B) = A^{2} - B^{2}$

(ii) The determinant of the product of the matrices is equal to product of their individual determinants, i.e. $| A B | = | A | | B |$ .

Given, $M^{2} = N^{4} \Rightarrow M^{2} - N^{4} = 0$

$\Rightarrow (M - N^{2}) (M + N^{2}) = 0$

[as $M N = N M$ ]

Also, $M \neq N^{2}$

$\Rightarrow M + N^{2} = 0$

$\Rightarrow \det (M + N^{2}) = 0$

Also, $\det (M^{2} + M N^{2}) = (\det M) (\det M + N^{2})$

$= (\det M) (0) = 0$

As, $\det (M^{2} + M N^{2}) = 0$

Thus, there exists a non-zero matrix $U$ such that

$(M^{2} + M N^{2}) U = 0$

पिछला

अगला

Matrices and Determinants 2 Question 22

24. Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M$ . Further, if $M \neq N^{2}$ and $M^{2} = N^{4}$ , then

Answer:

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Matrices and Determinants 2 Question 22

24. Let M and N be two 3×3 matrices such that MN=NM. Further, if M≠N2 and M2=N4, then

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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24. Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M$ . Further, if $M \neq N^{2}$ and $M^{2} = N^{4}$ , then