29. Let \[ A=\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 1\end{array}\right] \] We wish to determine all \( 4 \times 3 \) matrices \( X \) for which \( A X A=A \) Observe that \[ P A Q=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] \] where \( P \) and \( Q \) are the invertible matrices \[ P=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 2 & 1\end{array}\right] \text { and } Q=\left[\begin{array}{rrrr}1 & -1 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1\end{array}\right] . \] (a) Explain how you can use this to convert the problem of finding all \( X \) so that \( A X A=A \) to finding all \( 4 \times 3 \) matrices \( Y \) so that \( B Y B=B \), where \( B=P A Q \). Make sure to indicate how \( X \) and \( Y \) are related. (b) Find all \( Y \) so that \( B Y B=B \). (c) Use your answer in (b) to find all \( X \) so that \( A X A=A \). (d) Give 2 explicit \( X \) with \( A X A=A \).

d. What is the \( z \)-score for a music artist that uses 6,400 unique words in their songs?

28 In triunghiul \( A B C \) isoscel \( (A B=A C), B M \) și \( C N \) sunt mediane, \( M \in A C, N \in A B \), iar \( B M \cap C N=\{G\} \). Dacă \( P \) este mijlocul segmentului \( B G \), demonstrați că \( N P \perp B C \). 29 In paralelogramul \( A B C D \) se notează cu \( G \) și \( G^{\prime} \) centrele de greutate ale triunghiurilor \( A B D \) și \( B C D \). Arătați că patrulaterul \( B G D G^{\prime} \) este paralelogram. 30 In triunghiul \( A B C \) se cunosc \( A B=5 \mathrm{~cm} \) și \( B C=8 \mathrm{~cm} \). AM este mediană, \( M \in B C \), iar \( N \) este mijlocul lui \( A M \). Dacă \( B N \cap A C=\{P\} \) și \( A P=2 \mathrm{~cm} \), calculați perimetrul triunghiului \( A B C \).

1. Dalam sebuah kelas terdapat 9 siswa dan berikut adalah data nilai ujian matema \( 70,70,75,80,85,90,95,100 \). Tentukan nilai median dari data tersebut.

Joey is standing on a diving board that is 5.6 feet above sea level. Joey dives into the water to a depth of -8.6 feet below sea level. Which equation can be used to determine Joey's change in altitude during the dive? (1) \( 5.6-(-8.6)=14.2 \) (i) \( 5.6+3.0=8.6 \) (i) \( -5.6+14.2=-8.6 \) (i) \( 5.6-8.6=-14.2 \)

(b) Con los resultados obtenidos en el inciso anterior, calcular la derivada de las siguientes funciones: \( \begin{array}{ll}\text { (i) } g(x)=\tanh x & \text { (ii) } g(x)=\operatorname{sech}(x) \\ \text { (iii) } g(x)=\operatorname{cotanh} x & \text { (iv) } g(x)=\operatorname{cosech}(x)\end{array} \)