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If X and Y are n × n matrices with real entries, then which of the following is/are TRUE ?
- a)If P–1XP is diagonal for some real invertible matrix P, then there exists a basis for Rn consisting of eigenvectors of X.
- b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.
- c)If X2 is diagonal, then X is diagonal.
- d)If X is a diagonal and XY = YX for all Y, then X = λl for some λ ∈ R
Correct answer is option 'A,B,D'. Can you explain this answer?
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If X and Y are n n matrices with real entries, then which of the foll...
Statement a: If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rn consisting of eigenvectors of X.
Explanation:
To prove this statement, we need to show that if P1XP is diagonal, then X has a basis consisting of eigenvectors.
Let's assume that P1XP is diagonal, which means that there exists a diagonal matrix D such that P1XP = D.
Now, let's consider the equation Xv = λv, where v is a non-zero vector and λ is the eigenvalue corresponding to v.
We can rewrite this equation as (P1XP)v = λv.
Multiplying both sides by P-1, we get P-1(P1XP)v = P-1(λv), which simplifies to XPv = P-1(λv).
Since P-1 is invertible, we can rewrite the equation as XPv = μv, where μ = P-1(λv).
Now, let's consider the vector w = Pv.
Multiplying both sides of the equation by P, we get PW = PXPv = XP(Pv) = Xw.
Therefore, w is an eigenvector of X corresponding to the eigenvalue μ.
Since w = Pv and v is a non-zero vector, we can conclude that X has a basis consisting of eigenvectors.
Hence, statement a is true.
Statement b: If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.
Explanation:
To prove this statement, we need to show that if X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.
Let's assume that X is diagonal with distinct diagonal entries and XY = YX.
Let's consider the i-th column of XY. It can be written as (XY)i = X(Yi), where Yi is the i-th column of Y.
Since X is diagonal, the i-th column of XY is given by (XY)i = XiiYi, where Xii is the i-th diagonal entry of X.
Similarly, the i-th column of YX is given by (YX)i = YiiXi, where Yii is the i-th diagonal entry of Y.
Since XY = YX, we have XiiYi = YiiXi for all i.
Since Xii and Yii are distinct, we can conclude that Yi = 0 for all i.
Therefore, Y is diagonal with all diagonal entries being 0.
Hence, statement b is true.
Statement d: If X is a diagonal and XY = YX for all Y, then X = λ for some λ ∈ R.
Explanation:
To prove this statement, we need to show that if X is a diagonal matrix and XY = YX for all matrices Y, then all the diagonal entries of X are equal.
Let X = [x1 0 ... 0
0 x2 ... 0
. . .
. . .
. . .
0 0 ... xn]
Let Y = [y1 y2 ... yn
z1 z2 ...
Explanation:
To prove this statement, we need to show that if P1XP is diagonal, then X has a basis consisting of eigenvectors.
Let's assume that P1XP is diagonal, which means that there exists a diagonal matrix D such that P1XP = D.
Now, let's consider the equation Xv = λv, where v is a non-zero vector and λ is the eigenvalue corresponding to v.
We can rewrite this equation as (P1XP)v = λv.
Multiplying both sides by P-1, we get P-1(P1XP)v = P-1(λv), which simplifies to XPv = P-1(λv).
Since P-1 is invertible, we can rewrite the equation as XPv = μv, where μ = P-1(λv).
Now, let's consider the vector w = Pv.
Multiplying both sides of the equation by P, we get PW = PXPv = XP(Pv) = Xw.
Therefore, w is an eigenvector of X corresponding to the eigenvalue μ.
Since w = Pv and v is a non-zero vector, we can conclude that X has a basis consisting of eigenvectors.
Hence, statement a is true.
Statement b: If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.
Explanation:
To prove this statement, we need to show that if X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.
Let's assume that X is diagonal with distinct diagonal entries and XY = YX.
Let's consider the i-th column of XY. It can be written as (XY)i = X(Yi), where Yi is the i-th column of Y.
Since X is diagonal, the i-th column of XY is given by (XY)i = XiiYi, where Xii is the i-th diagonal entry of X.
Similarly, the i-th column of YX is given by (YX)i = YiiXi, where Yii is the i-th diagonal entry of Y.
Since XY = YX, we have XiiYi = YiiXi for all i.
Since Xii and Yii are distinct, we can conclude that Yi = 0 for all i.
Therefore, Y is diagonal with all diagonal entries being 0.
Hence, statement b is true.
Statement d: If X is a diagonal and XY = YX for all Y, then X = λ for some λ ∈ R.
Explanation:
To prove this statement, we need to show that if X is a diagonal matrix and XY = YX for all matrices Y, then all the diagonal entries of X are equal.
Let X = [x1 0 ... 0
0 x2 ... 0
. . .
. . .
. . .
0 0 ... xn]
Let Y = [y1 y2 ... yn
z1 z2 ...
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If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer?
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If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer?.
If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer?.
Solutions for If X and Y are n n matrices with real entries, then which of the following is/are TRUE ?a)If P1XP is diagonal for some real invertible matrix P, then there exists a basis for Rnconsisting of eigenvectors of X.b)If X is diagonal with distinct diagonal entries and XY = YX, then Y is also diagonal.c)If X2 is diagonal, then X is diagonal.d)If X is a diagonal and XY = YX for all Y, then X = lfor some RCorrect answer is option 'A,B,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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