Mathematics Exam > Mathematics Questions > Let P be a matrix of size 3 x 3 with eigenval...
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Let P be a matrix of size 3 x 3 with eigenvalues 1,2 and 3. Then P is
- a)neither invertible nor diagonalizable
- b)both invertible and diagonalizable
- c)invertible but not diagonalizable
- d)not invertible but diagonalizable
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let P be a matrix of size 3 x 3 with eigenvalues 1,2 and 3. Then P isa...
Since, all the eigenvalues are distinct, the matrix will be similar to diagonal matrix and hence diagonalizable. Since, none of the eigenvalue is zero. So, P is invertible.
Most Upvoted Answer
Let P be a matrix of size 3 x 3 with eigenvalues 1,2 and 3. Then P isa...
Solution:
Given:
- Matrix P is of size 3 x 3
- Eigenvalues of P are 1, 2, and 3
To determine the properties of matrix P, we need to understand the concepts of invertibility and diagonalizability.
Invertibility:
A matrix is invertible if its determinant is non-zero. The determinant of a matrix is the product of its eigenvalues. Therefore, if a matrix has a non-zero determinant, it means that none of its eigenvalues are zero.
In this case, the eigenvalues of matrix P are 1, 2, and 3. Since none of these eigenvalues are zero, the determinant of P is non-zero. Hence, matrix P is invertible.
Diagonalizability:
A matrix is diagonalizable if it can be expressed as a product of three matrices: P = SΛS^(-1), where Λ is a diagonal matrix and S is an invertible matrix consisting of the eigenvectors of P.
To check if matrix P is diagonalizable, we need to ensure that it has a complete set of linearly independent eigenvectors.
Since P is a 3 x 3 matrix and has three distinct eigenvalues (1, 2, and 3), it implies that P has three linearly independent eigenvectors. Therefore, matrix P is diagonalizable.
Conclusion:
Based on the above analysis, we can conclude that matrix P is both invertible and diagonalizable, as mentioned in option 'B'.
Given:
- Matrix P is of size 3 x 3
- Eigenvalues of P are 1, 2, and 3
To determine the properties of matrix P, we need to understand the concepts of invertibility and diagonalizability.
Invertibility:
A matrix is invertible if its determinant is non-zero. The determinant of a matrix is the product of its eigenvalues. Therefore, if a matrix has a non-zero determinant, it means that none of its eigenvalues are zero.
In this case, the eigenvalues of matrix P are 1, 2, and 3. Since none of these eigenvalues are zero, the determinant of P is non-zero. Hence, matrix P is invertible.
Diagonalizability:
A matrix is diagonalizable if it can be expressed as a product of three matrices: P = SΛS^(-1), where Λ is a diagonal matrix and S is an invertible matrix consisting of the eigenvectors of P.
To check if matrix P is diagonalizable, we need to ensure that it has a complete set of linearly independent eigenvectors.
Since P is a 3 x 3 matrix and has three distinct eigenvalues (1, 2, and 3), it implies that P has three linearly independent eigenvectors. Therefore, matrix P is diagonalizable.
Conclusion:
Based on the above analysis, we can conclude that matrix P is both invertible and diagonalizable, as mentioned in option 'B'.
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