Chapter 3: Q5P (page 112)
In Problems 1 to 5, all lines are in the plane.
5. Write, in parametric form, the equation of the axis.
Short Answer
The parametric form of the equation of - axis is .
Chapter 3: Q5P (page 112)
In Problems 1 to 5, all lines are in the plane.
5. Write, in parametric form, the equation of the axis.
The parametric form of the equation of - axis is .
All the tools & learning materials you need for study success - in one app.
Get started for freeEvaluate the determinants in Problems 1 to 6 by the methods shown in Example 4. Remember that the reason for doing this is not just to get the answer (your computer can give you that) but to learn how to manipulate determinants correctly. Check your answers by computer.
Draw diagrams and prove (4.1).
Show that an orthogonal matrix M with all real eigenvalues is symmetric. Hints: Method 1. When the eigenvalues are real, so are the eigenvectors, and the unitary matrix which diagonalizes M is orthogonal. Use (11.27). Method 2. From Problem 46, note that the only real eigenvalues of an orthogonal M are ±1. Thus show that . Remember that M is orthogonal to show that .
Show that ifA and Bare matrices which don't commute, then , but if they do commute then the relation holds. Hint: Write out several terms of the infinite series for , and and, do the multiplications carefully assuming that anddon't commute. Then see what happens if they do commute
The Caley-Hamilton theorem states that "A matrix satisfies its own characteristic equation." Verify this theorem for the matrix in equation (11.1). Hint: Substitute the matrixforrole="math" localid="1658822242352" in the characteristic equation (11.4) and verify that you have a correct matrix equation. Further hint: Don't do all the arithmetic. Use (11.36) to write the left side of your equation asand show that the parenthesis. Remember that, by definition, the eigenvalues satisfy the characteristic equation.
What do you think about this solution?
We value your feedback to improve our textbook solutions.