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Chapter 3: Q8P (page 95)
Show that if, in using the Laplace development, you accidentally multiply the elements of one row by the cofactors of another row, you get zero. Hint: Consider Fact 2b
It has been proved that zero is obtained.
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Find the symmetric equations and the parametric equations of a line, and/or the equation of the plane satisfying the following given conditions.
Line through and parallel to the line
Compute the product of each of the matrices in Problem 4with its transpose [see (2.2)or (9.1)in both orders, that isand, etc.
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.
Show that a real Hermitian matrix is symmetric. Show that a real unitary matrix is orthogonal. Note: Thus, we see that Hermitian is the complex analogue of symmetric, and unitary is the complex analogue of orthogonal. (See Section 11.)
Find the Eigen values and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer.
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