Mathematical Immaturity

2.16 Exercises

• If $A = \begin{bmatrix} 1 & -4 & -2 \\ -1 & 4 & -2 \end{bmatrix},$ $B = \begin{bmatrix} 1 & 2 \\ -1 & 3 \\ 5 & -2 \end{bmatrix},$ $C = \begin{bmatrix} 2 & 2 \\ 1 & -1 \\ 1 & -3 \end{bmatrix},$ compute $B + C,$ $AB,$ $BA,$ $AC,$ $CA,$ $A(2B - 3C).$
• Let $A = \begin{bmatrix} 0 & 1 \\ 0 & 2 \end{bmatrix}.$ Find all 2 x 2 matrices $B$ such that (a) $AB = O$; (b) $BA = O.$
• In each case find $a,$ $b,$ $c,$ $d$ to satisfy the given equation.
  
  $\quad$ (a) $\quad \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} = \begin{bmatrix} 1 \\ 9 \\ 6 \\ 5 \end{bmatrix};$
  
  $\quad$ (b) $\quad \begin{bmatrix} a & b & c \\ 1 & 4 & 9 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 6 & 6 \\ 1 & 9 & 8 & 4 \end{bmatrix}.$
• Calculate $AB - BA$ in each case.
  
  $\quad$ (a) $\quad A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \end{bmatrix},$ $\quad B = \begin{bmatrix} 4 & 1 & 1 \\ -4 & 2 & 0 \\ 1 & 2 & 1 \end{bmatrix};$
  
  $\quad$ (b) $\quad A = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 1 & 2 \\ -1 & 2 & 1 \end{bmatrix},$ $\quad B = \begin{bmatrix} 3 & 1 & -2 \\ 3 & -2 & 4 \\ -3 & 5 & 11 \end{bmatrix}.$
• If $A$ is a square matrix, prove that $A^n A^m = A^{m+n}$ for all integers $m \geq 0,$ $n \geq 0.$
• Let $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.$ Verify that $A^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and compute $A^n.$
• Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}.$ Verify that $A^2 = \begin{bmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{bmatrix}$ and compute $A^n.$
• Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}.$ Verify that $A^2 = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}.$ Compute $A^3$ and $A^4.$ Guess a general formula for $A^n$ and prove it by induction.
• Let $A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}.$ Prove that $A^2 = 2A - I$ and compute $A^{100}.$
• Find all 2 x 2 matrices $A$ such that $A^2 = O.$
• (a) Prove that a 2 x 2 matrix $A$ commutes with every 2 x 2 matrix if and only if $A$ commutes with each of the four matrices $$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.$$
  (b) Find all such matrices $A.$
• The equation $A^2 = I$ is satisfied by each of the 2 x 2 matrices $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \begin{bmatrix} 1 & 0 \\ c & -1 \end{bmatrix}, \quad \begin{bmatrix} 1 & b \\ 0 & -1 \end{bmatrix},$$ where $b$ and $c$ are arbitrary real numbers. Find all 2 x 2 matrices $A$ such that $A^2 = I.$
• If $A = \begin{bmatrix} 2 & -1 \\ -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 7 & 6 \\ 9 & 8 \end{bmatrix},$ find 2 x 2 matrices $C$ and $D$ such that $AC = B$ and $DA = B.$
• (a) Verify that the algebraic identities $$(A + B)^2 = A^2 + 2AB + B^2 \quad \text{and} \quad (A + B)(A - B) = A^2 - B^2$$ do not hold for the 2 x 2 matrices $A = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix}.$
  (b) $\quad$ Amend the right-hand members of these identities to obtain formulas valid for all square matrices $A$ and $B.$
  (c) $\quad$ For which matrices $A$ and $B$ are the identities valid as stated in (a)?