7 Linear Algebra IV

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. For the matrix of the last question of the previous homework. find one row or column (state which row or column you took) of the inverse matrix using minors. Select the row or column so that you can reuse your earlier results, if correct, or else the posted solution, to cut down on the work.

2. For the matrix
   
   $$A = \left( \begin{array}{rrr} 11 & 0 & -5 \\ 0 & 1 & 0 \\ 4 & -7 & 9 \end{array} \right)$$
   
   
   find the inverse both using minors and using Gaussian elimination. Make sure the result is the same.

3. For the matrix
   
   $$A = \left( \begin{array}{rrr} 6 & -2 \\ -3 & 4 \end{array} \right)$$
   
   
   find the eigenvalues and eigenvectors. Take $\lambda_1$ to be the larger of the eigenvalues and $\lambda_2$ the smaller one. Is the matrix defective? Singular? What is the rank?

4. Take the two eigenvectors of the previous problem to be $(2,1-\sqrt7)$ and $(2,1+\sqrt7)$. (The eigenvectors that you found may have been different by some scalar factor.) Suppose you use the given two eigenvectors now as the basis of a new coordinate system. How do the coordinates $v_1,v_2$ of a vector $\vec v$ in the old coordinate system relate to the ones $v_1',v_2'$ in the new coordinate system and vice versa? The instructor said that matrix $A'=P^{-1}AP$ in the new coordinates is a diagonal matrix. Verify that by direct matrix multiplication. Are the eigenvalues on the main diagonal? In the same order as the corresponding eigenvectors in $P$?

5. For the matrix
   
   $$A = \left( \begin{array}{rrrr} -2 & 1 & 0 & 0 \\ 1 & ... ... 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)$$
   
   
   find the eigenvalues and eigenvectors. Is the matrix defective? Singular? What is the rank?

6. Here are some quick ones. For each answer, explain why.
   1. If a matrix is singular, how does that reflect in its eigenvalues?
   2. Is it possible for an $n\times n$ matrix with all $n$ eigenvalues different to be defective?
   3. Is a square null-matrix defective? Singular?
   4. Is a unit matrix defective? Singular?
   5. Is a square matrix with all coefficients 1 singular? Defective?
   6. Is a square matrix with all coefficients 0 except $a_{12}=1$ singular? Defective? How many independent eigenvectors are there? To answer this, find the eigenvalue(s) and dimension of their null spaces for this simple triangular matrix.
   7. Is a square matrix with all coefficients 0 except $a_{i i+1}=1$ for $i=1,2,\ldots,n-1$ singular? Defective? How many independent eigenvectors are there?