Practice Statement (21.12.1) - Linear Algebra - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
What is the Cayley-Hamilton theorem?
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Think about matrices and their equations.
Question 2.
Can a non-singular matrix be inverted?
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Recall the properties of determinants.
3Question 3.
What does the Cayley-Hamilton theorem state?
- Every matrix can be inverted
- Every square matrix satisfies its own characteristic polynomial
- Every matrix has a determinant
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Think about any distinctions between types of matrices.
4Question 4.
True or False: All matrices satisfy the Cayley-Hamilton theorem.
- True
- False
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Consider the definition of square matrices.
Question 5.
Given the matrix A = [[4, 1], [2, 3]], calculate its characteristic polynomial, apply the Cayley-Hamilton theorem, and find A^{-1} with the theorem's help.
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Utilize the determinant and characteristics from previous examples.
Question 6.
Using the Cayley-Hamilton theorem, express A^4 in terms of A^3 and A^2 for the matrix A = [[1, 2], [0, 1]].
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Consider the pattern formed in the powers of A to simplify.

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