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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2: Did you read my tips on how to study and learn this course? If not, here is a link to those important suggestions: The department posted SOLUTIONS for Assignment 1 (these are not my solutions). Click
here.Here is a link to the actual assignment, for those of you who don't have it: Study Lesson 3 (Matrix Math), Lesson 4 (The Inverse of a Matrix and Applications), Lesson 5 (Elementary Matrices), and Lesson 6 (Determinants and Their Properties)
from my Linear Algebra & Vector Geometry book to prepare for this assignment.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test. Don't let that unusual symbol confuse you. That is just an unknown constant scalar λ. Just take this question slow. First, multiply the Identity matrix I by λ. That gives you a diagonal matrix with λ the entry down the main diagonal. Then, compute A minus
that diagonal matrix. Then compute the determinant of that 2x2 matrix and simplify, and you end up with a λ^2 polynomial. Part (a) is now done. You know have the P(λ) polynomial he has requested. Then, just sub the matrix A in place of λ to do part (b). As he warns, if you have done it correctly, you will be looking at a matrix math problem where you are, among other things, being told to square matrix A, so make sure you square
that properly! That is A times A. Don't you dare just square every number in A! But, you can't just add or subtract a scalar value, like 4, for example, from a matrix. You must realize that you are actually being asked to subtract that scalar times I, an Identity matrix. That's what he means by "+ cI".
Hint: If you have done everything right in part (b),
your answer will be a zero matrix.
Use the Properties of Inverse Matrices I list in Lesson 3, page 86. Make sure you get rid of all the brackets.
Use the Equivalent Statements Theorem, as listed on page 189 in Lesson 6. Note that two matrices are row equivalent if one can be derived from the other by elementary row operations. So, every matrix that you generate each
step of the way during a row reduction process is row equivalent to every other matrix in that row reduction process. I could restate part (c) of that equivalent statements theorem to say that A is row equivalent to I, the identity matrix. In other words, if we know A is an invertible matrix (second statement in the equivalent statements theorem), then it is also true that A is row equivalent to I. And if B is row equivalent to A, what does that
mean?
This one is really weird. He is telling you that you are only allowed to put 0 or 1 in place of the entries of a 3x3 matrix A. I suggest you first set up a general 3x3 matrix and compute it for 9 unknowns, similar to what I do when demonstrating how to compute a 3x3
determinant at the bottom of page 203 in my book. Now, use trial and error, replacing each unknown with either 0 or 1 to see the biggest possible answer you can make and the smallest possible answer you can make. Hint: You can get a larger answer than 1, and a smaller answer than -1.
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