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For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?
- a)|FE| = |I - FE| |FGE|
- b)(I - FE)(I + FGE) = I
- c)EFG = GEF
- d)(I - FE)(I - FGE) = I
Correct answer is option 'A,B,C'. Can you explain this answer?
Most Upvoted Answer
For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be ...
Solution:
Given,
- |M| represents the determinant of the matrix M.
- I is the 3 x 3 identity matrix.
- E and F are two 3 x 3 matrices such that (I - EF) is invertible.
- G = (I - EF)-1.
We need to determine which of the given statements are true.
a) |FE| = |I - FE| |FGE|
To prove this statement, we need to use the following properties of determinants:
- |AB| = |A| |B| (Property 1)
- |A-1| = 1/|A| (Property 2)
Now, we have:
|I - EF| = |(I - EF)G(I - EF)-1| (Multiplying both sides with G and G-1)
|I - EF| = |G(I - EF)(I - EF)-1| (Rearranging terms)
|I - EF| = |G| |(I - EF)(I - EF)-1| (Using Property 1)
|I - EF| = |G| (1/|I - EF|) (Using Property 2)
|I - EF|2 = |G|
Also, we have:
|FGE| = |F| |G| |E| (Using Property 1)
Therefore,
|FE| = |F| |E|
|I - FE| = |I| |I - EF| (Using Property 1)
|I - FE| = |I - EF|
|FGE| = |F| |G| |E| (Using Property 1)
Now, substituting the above values in the given equation, we get:
|FE| = |I - FE| |FGE|
|F| |E| = |I - EF| |F| |G| |E|
|I - EF|2 |G| = |I - EF| |G| |F| |E|
|I - EF| |G| = |F| |E|
Substituting the value of |I - EF| |G| from above, we get:
|FE| = |I - FE| |FGE|
|F| |E| = (|I - EF| |G|) |F| |E|
|F| |E| = |F| |E|
Hence, the statement a) is true.
b) (I - FE)(I + FGE) = I
To prove this statement, we need to use the following property of matrices:
- (AB)-1 = B-1 A-1 (Property 3)
Now, we have:
(I - EF)(I + FGE) = I - EF + FGE - EFGE
(I - EF)(I + FGE)(I - EF)-1 = (I - EF + FGE - EFGE)(I - EF)-1
(I - EF)(I + FGE)(I - EF)-1 = I - EF + FGE (Using Property 3)
(I - EF)(I + FGE)(I - EF)-1 = I + (I - EF)G (Substituting G = (I - EF)-1)
(I - EF)(I + FGE)(I - EF)-
Given,
- |M| represents the determinant of the matrix M.
- I is the 3 x 3 identity matrix.
- E and F are two 3 x 3 matrices such that (I - EF) is invertible.
- G = (I - EF)-1.
We need to determine which of the given statements are true.
a) |FE| = |I - FE| |FGE|
To prove this statement, we need to use the following properties of determinants:
- |AB| = |A| |B| (Property 1)
- |A-1| = 1/|A| (Property 2)
Now, we have:
|I - EF| = |(I - EF)G(I - EF)-1| (Multiplying both sides with G and G-1)
|I - EF| = |G(I - EF)(I - EF)-1| (Rearranging terms)
|I - EF| = |G| |(I - EF)(I - EF)-1| (Using Property 1)
|I - EF| = |G| (1/|I - EF|) (Using Property 2)
|I - EF|2 = |G|
Also, we have:
|FGE| = |F| |G| |E| (Using Property 1)
Therefore,
|FE| = |F| |E|
|I - FE| = |I| |I - EF| (Using Property 1)
|I - FE| = |I - EF|
|FGE| = |F| |G| |E| (Using Property 1)
Now, substituting the above values in the given equation, we get:
|FE| = |I - FE| |FGE|
|F| |E| = |I - EF| |F| |G| |E|
|I - EF|2 |G| = |I - EF| |G| |F| |E|
|I - EF| |G| = |F| |E|
Substituting the value of |I - EF| |G| from above, we get:
|FE| = |I - FE| |FGE|
|F| |E| = (|I - EF| |G|) |F| |E|
|F| |E| = |F| |E|
Hence, the statement a) is true.
b) (I - FE)(I + FGE) = I
To prove this statement, we need to use the following property of matrices:
- (AB)-1 = B-1 A-1 (Property 3)
Now, we have:
(I - EF)(I + FGE) = I - EF + FGE - EFGE
(I - EF)(I + FGE)(I - EF)-1 = (I - EF + FGE - EFGE)(I - EF)-1
(I - EF)(I + FGE)(I - EF)-1 = I - EF + FGE (Using Property 3)
(I - EF)(I + FGE)(I - EF)-1 = I + (I - EF)G (Substituting G = (I - EF)-1)
(I - EF)(I + FGE)(I - EF)-
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Community Answer
For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be ...
G(I - EF) = (I - EF)G = I
⇒ G - GEF = G - EFG = I ... (1)
(1) |FE| = |I - FE| |FGE| = |FGE - FEFGE|
= |FGE - F(G - I)E| = |FGE - FGE + FE| = |FE|
(2) (I - FE)(I + FGE) = I + FGE - FE - FEFGH
= I + FGE - FE - F(G - I)E = I + FGE - FE - FGE + FE = I
(3) From (I), it is true.
(4) (I - FE)(I - FGE) = I - FGE - FE + FEFGE
= I - FGE - FE + F(G - I)E = I - FGE - FE + FGE - FE
= I - 2FE
⇒ G - GEF = G - EFG = I ... (1)
(1) |FE| = |I - FE| |FGE| = |FGE - FEFGE|
= |FGE - F(G - I)E| = |FGE - FGE + FE| = |FE|
(2) (I - FE)(I + FGE) = I + FGE - FE - FEFGH
= I + FGE - FE - F(G - I)E = I + FGE - FE - FGE + FE = I
(3) From (I), it is true.
(4) (I - FE)(I - FGE) = I - FGE - FE + FEFGE
= I - FGE - FE + F(G - I)E = I - FGE - FE + FGE - FE
= I - 2FE
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For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer?
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For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer?.
For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer?.
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For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer?, a detailed solution for For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? has been provided alongside types of For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice For any 3 x 3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3 x 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is/are TRUE?a)|FE| = |I - FE| |FGE|b)(I - FE)(I + FGE) = Ic)EFG = GEFd)(I - FE)(I - FGE) = ICorrect answer is option 'A,B,C'. Can you explain this answer? tests, examples and also practice JEE tests.
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