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The product of matrix A and adj. A is a
- a)zero matrix
- b)unit matrix
- c)scalar matrix
- d)matrix A itself
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The product of matrix A and adj. A is aa)zero matrixb)unit matrixc)sc...
Understanding the Product of Matrix A and its Adjugate
The relationship between a matrix and its adjugate is fundamental in linear algebra. Let's explore why the product of matrix A and its adjugate (adj A) is a scalar matrix.
Definition of Adjugate
- The adjugate (or adjoint) of a matrix A is the transpose of its cofactor matrix.
- For an n x n matrix, the adjugate plays a crucial role in finding the inverse.
Key Property
- The key property to remember is that:
- A * adj A = det(A) * I_n
where I_n is the identity matrix of size n x n, and det(A) is the determinant of A.
When is the Product a Scalar Matrix?
- If the determinant of A (det(A)) is a scalar value (which it always is), the product A * adj A results in a scalar matrix.
- A scalar matrix is a matrix that is a constant multiple of the identity matrix.
Conclusion
- Therefore, if A is a non-zero matrix, the product A * adj A yields:
- A scalar multiple of the identity matrix, indicating that the correct answer is option C: a scalar matrix.
Importance in Applications
- This relationship is essential in various applications, including solving systems of linear equations, finding inverses, and understanding linear transformations.
By grasping this concept, you can enhance your understanding of matrix operations in mechanical engineering and other fields.
The relationship between a matrix and its adjugate is fundamental in linear algebra. Let's explore why the product of matrix A and its adjugate (adj A) is a scalar matrix.
Definition of Adjugate
- The adjugate (or adjoint) of a matrix A is the transpose of its cofactor matrix.
- For an n x n matrix, the adjugate plays a crucial role in finding the inverse.
Key Property
- The key property to remember is that:
- A * adj A = det(A) * I_n
where I_n is the identity matrix of size n x n, and det(A) is the determinant of A.
When is the Product a Scalar Matrix?
- If the determinant of A (det(A)) is a scalar value (which it always is), the product A * adj A results in a scalar matrix.
- A scalar matrix is a matrix that is a constant multiple of the identity matrix.
Conclusion
- Therefore, if A is a non-zero matrix, the product A * adj A yields:
- A scalar multiple of the identity matrix, indicating that the correct answer is option C: a scalar matrix.
Importance in Applications
- This relationship is essential in various applications, including solving systems of linear equations, finding inverses, and understanding linear transformations.
By grasping this concept, you can enhance your understanding of matrix operations in mechanical engineering and other fields.
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Community Answer
The product of matrix A and adj. A is aa)zero matrixb)unit matrixc)sc...
The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinate.
Consider A be a square matrix, then
(Adjoint A)・A = A ・(Adjoint A) = , A , ・I
=,A, I
Which is the scalar matrix
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The product of matrix A and adj. A is aa)zero matrixb)unit matrixc)scalar matrixd)matrix A itselfCorrect answer is option 'C'. Can you explain this answer?
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