Finding Characteristic Polynomial of a Matrix

Finding the Characteristic Polynomial

• The characteristic polynomial of a matrix is found by calculating the determinant of the matrix (A - λI), where λ is a scalar and I is the identity matrix.
• Step 1: Subtract λ from the diagonal elements of the matrix A.
• Step 2: Calculate the determinant of the resulting matrix.
• Step 3: The determinant will be a polynomial in λ, which is the characteristic polynomial.
• The characteristic polynomial of the 3x3 matrix A will be a cubic polynomial.
• The characteristic polynomial of A is written as:
  • $P\lambda^3 + Q\lambda^2 + R\lambda + S = 0$
  • Where P, Q, R, and S are constants that depend on the elements of the matrix A.

Solving for the Coefficients

• P is the coefficient of the λ³ term, which is always 1.
• Q is the coefficient of the λ² term.
• R is the coefficient of the λ term.
• S is the constant term.

The Correct Statements

• (a) P = a - e + i This statement is incorrect. The coefficient of the λ³ term is always 1, regardless of the values of the elements in the matrix.
• (b) Q = a + e + i This statement is incorrect. Q is the sum of the diagonal elements and is not equal to a + e + i.
• (c) Q = ae - ie + ia This statement is incorrect. Q is the sum of the diagonal elements, not their product or any combination thereof.
• (d) P = 0 This statement is incorrect. The coefficient of the λ³ term is always 1.
• (e) none of these This statement is correct. None of statements (a) to (d) are true.
• The correct expression for Q would be the sum of the diagonal elements of the matrix: a + e + i.

Key Points

• The characteristic polynomial of a matrix is a critical tool in linear algebra.
• It helps determine the eigenvalues of a matrix, which in turn provide information about the matrix's behavior in relation to linear transformations.
• The coefficients of the characteristic polynomial are important for understanding the properties of the matrix.

This quiz covers the steps involved in finding the characteristic polynomial of a matrix. You'll learn how to calculate the determinant of (A - λI) and identify the coefficients of the resulting cubic polynomial. Test your understanding of this fundamental concept in linear algebra.