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Let y be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?
- a)yyT is symmetric matrix
- b)yyT is invertible
- c)yyT has a rank of 2022
- d)yTy is an eigen value of yyT.
Correct answer is option 'A,D'. Can you explain this answer?
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Let y be a non-zero vector of size 2022 x 1. Which of the following st...
Statement: Let y be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?
a) yyT is a symmetric matrix
b) yyT is invertible
c) yyT has a rank of 2022
d) yTy is an eigenvalue of yyT.
Explanation:
Statement a) yyT is a symmetric matrix:
- The product of a vector y with its transpose yyT will result in a square matrix.
- A matrix is symmetric if it is equal to its transpose, i.e., A = A^T.
- In this case, yyT is a symmetric matrix because (yyT)^T = (y^T)^T * y^T = yyT.
Statement d) yTy is an eigenvalue of yyT:
- The eigenvalues of a matrix A are the values λ for which there exists a non-zero vector x, such that Ax = λx.
- Let's consider the product yTy. This is a scalar value.
- Now, let's consider the product yyT. This is a square matrix.
- If we find the eigenvalues of yyT, they will be the same as the eigenvalues of yTy because yTy is a scalar value and can be considered as a 1x1 matrix.
- Therefore, yTy is an eigenvalue of yyT.
Statement b) yyT is invertible:
- A matrix is invertible if it has a non-zero determinant.
- The determinant of yyT is the determinant of y multiplied by the determinant of y transpose.
- Since y is a non-zero vector, its determinant is non-zero.
- The determinant of a matrix and its transpose are equal.
- Therefore, the determinant of yyT is non-zero, making yyT invertible.
Statement c) yyT has a rank of 2022:
- The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
- The rank of a product of two matrices is equal to the rank of either matrix, at most.
- In this case, the rank of yyT is equal to the rank of y, which is 1 because y is a non-zero vector.
- Therefore, yyT has a rank of 1, not 2022.
Conclusion:
- The correct statements are a) yyT is a symmetric matrix and d) yTy is an eigenvalue of yyT.
- The statement b) yyT is invertible is incorrect.
- The statement c) yyT has a rank of 2022 is incorrect.
a) yyT is a symmetric matrix
b) yyT is invertible
c) yyT has a rank of 2022
d) yTy is an eigenvalue of yyT.
Explanation:
Statement a) yyT is a symmetric matrix:
- The product of a vector y with its transpose yyT will result in a square matrix.
- A matrix is symmetric if it is equal to its transpose, i.e., A = A^T.
- In this case, yyT is a symmetric matrix because (yyT)^T = (y^T)^T * y^T = yyT.
Statement d) yTy is an eigenvalue of yyT:
- The eigenvalues of a matrix A are the values λ for which there exists a non-zero vector x, such that Ax = λx.
- Let's consider the product yTy. This is a scalar value.
- Now, let's consider the product yyT. This is a square matrix.
- If we find the eigenvalues of yyT, they will be the same as the eigenvalues of yTy because yTy is a scalar value and can be considered as a 1x1 matrix.
- Therefore, yTy is an eigenvalue of yyT.
Statement b) yyT is invertible:
- A matrix is invertible if it has a non-zero determinant.
- The determinant of yyT is the determinant of y multiplied by the determinant of y transpose.
- Since y is a non-zero vector, its determinant is non-zero.
- The determinant of a matrix and its transpose are equal.
- Therefore, the determinant of yyT is non-zero, making yyT invertible.
Statement c) yyT has a rank of 2022:
- The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
- The rank of a product of two matrices is equal to the rank of either matrix, at most.
- In this case, the rank of yyT is equal to the rank of y, which is 1 because y is a non-zero vector.
- Therefore, yyT has a rank of 1, not 2022.
Conclusion:
- The correct statements are a) yyT is a symmetric matrix and d) yTy is an eigenvalue of yyT.
- The statement b) yyT is invertible is incorrect.
- The statement c) yyT has a rank of 2022 is incorrect.
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Community Answer
Let y be a non-zero vector of size 2022 x 1. Which of the following st...
Let,
Here, Rank of matrix ρ(y) = ρ(yT) = ρ(yyT) = ρ(yTy) = 1
Eigen value of yyT = 3, 0, 0
Here, yTy is eigen value of yyT i.e. 3.
Here, Rank of matrix ρ(y) = ρ(yT) = ρ(yyT) = ρ(yTy) = 1
Eigen value of yyT = 3, 0, 0
Here, yTy is eigen value of yyT i.e. 3.
- yyT is symmetrical matrix.
- ∵ Determinant of yyT is zero.
∴ A-1 will not exist. - Rank of yyT is 1.
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Let y be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?a)yyT is symmetric matrixb)yyT is invertiblec)yyT has a rank of 2022d)yTy is an eigen value of yyT.Correct answer is option 'A,D'. Can you explain this answer?
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