Full d'exercicis II

Exercici 1

Calcula els següents determinants:

a) \(\left\| \begin{array}{rr} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{array} \right\|\)	Solució:	\( 1 \)
b) \(\left\| \begin{array}{rrr} 0&1&1 \\ 1&1&0 \\ 1&0&1 \end{array} \right\|\)	Solució:	\( -2 \)
c) \(\left\| \begin{array}{rrr} 1&3&5 \\ 2&4&6 \\ 4&10&16 \end{array} \right\|\)	Solució:	\( 0 \)
d) \(\left\| \begin{array}{rrrr} 1&-5&7&0 \\ -3&0&1&0 \\ 4&-2&0&1 \\ 1&1&2&1 \end{array} \right\|\)	Solució:	\( 81 \)

Exercici 2

Resol les següents equacions:

a) \( \left\| \begin{array}{ccc} x-2&4&4 \\ 4&x-2&4 \\ 4&4&x-2 \end{array} \right\| = 0\)	Solució:	\( x=6 \quad \lor \quad x=-6 \)
b) \( \left\| \begin{array}{ccc} x&1&4 \\ 4&x&1 \\ 1&4&x \end{array} \right\| = 0\)	Solució:	\( x=-5 \)
c) \( \left\| \begin{array}{ccc} x+1&2&3 \\ 1&x+2&3 \\ 1&2&x+3 \end{array} \right\| = 0\)	Solució:	\( x=0 \quad \lor \quad x=-6 \)

Exercici 3

Resol les equacións

a) \( \left\| \begin{array}{ccc} x&1&0 \\ 1&x&1 \\ 0&1&x \end{array} \right\| = 0\)	Solució:	\( x=0 \quad \lor \quad x=\pm\sqrt 2 \)
b) \( \left\| \begin{array}{rrr} x&1&0 \\ x&1&-1 \\ 1&x&1 \end{array} \right\| = 0\)	Solució:	\( x=\pm 1 \)

Exercici 4

Si \(\boldsymbol{A}\) i \(\boldsymbol{B}\) són dues matrius quadrades d'ordre \(4\) que verifiquen que \(\left|\boldsymbol{A}\right|=6\) i \(\left|\boldsymbol{B}\right|=-5\), calcula:

a) \( \left\| \boldsymbol{B}^2 \right\|\)	Solució:	\( \left\| \boldsymbol{B}^2 \right\| = (-5)^2 = 25 \)
b) \( \left\| 5\boldsymbol{A} \right\|\)	Solució:	\( \left\| 5\boldsymbol{A} \right\| = 5^4 \cdot \left\| \boldsymbol{A} \right\| = 625 \cdot 6 = 3750 \)
c) \( \left\| -\boldsymbol{A} \right\|\)	Solució:	\( \left\| -\boldsymbol{A} \right\| = (-1)^4 \cdot \left\| \boldsymbol{A} \right\| = 1 \cdot 6 = 6 \)
d) \( \left\| \boldsymbol{A}^\mathsf{T} \cdot \boldsymbol{B} \right\|\)	Solució:	\( \left\| \boldsymbol{A}^\mathsf{T} \cdot \boldsymbol{B} \right\| = \left\| \boldsymbol{A}^\mathsf{T} \right\| \cdot \left\| \boldsymbol{B} \right\| = \left\| \boldsymbol{A} \right\| \cdot \left\| \boldsymbol{B} \right\| = 6 \cdot (-5) = -30 \)
e) \( \left\| \boldsymbol{B}^{-1} \right\|\)	Solució:	\( \left\| \boldsymbol{B}^{-1} \right\| = \displaystyle \frac{1}{\left\| \boldsymbol{B} \right\|} = -\frac{1}{5} \)

Exercici 5

Si sabem que \( \left|\boldsymbol{A}\right|=\left|\begin{array}{rr} a&b\\c&d \end{array}\right|=5 \), aplicant les propietats dels determinants, calcula:

a) \( \left\|\begin{array}{rr} 3a&3b\\c&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 3a&3b\\c&d \end{array}\right\| = 3·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 15 \)
b) \( \left\|\begin{array}{rr} -b&-a\\d&c \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} -b&-a\\d&c \end{array}\right\| = -1·\left\|\begin{array}{rr} b&a\\d&c \end{array}\right\| = \left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 5 \)
c) \( \left\|\begin{array}{rr} 4a&4b\\4c&4d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 4a&4b\\4c&4d \end{array}\right\| = 4^2·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 80 \)
d) \( \left\|\begin{array}{rr} -6a&-2b\\3c&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} -6a&-2b\\3c&d \end{array}\right\| = -2·\left\|\begin{array}{rr} 3a&b\\3c&d \end{array}\right\| = -2·3·5 = -30 \)
e) \( \left\|\begin{array}{rr} 3a-2b&b\\3c-2d&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 3a-2b&b\\3c-2d&d \end{array}\right\| = \left\|\begin{array}{rr} 3a&b\\3c&d \end{array}\right\| = 3·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 15 \)
f) \( \left\| \boldsymbol{A}^{-1} \right\| \)	Solució:	\( \displaystyle \left\|\boldsymbol{A}^{-1}\right\| = \frac{1}{\left\|\boldsymbol{A}\right\|} = \frac{1}{5} \)
g) \( \left\| \boldsymbol{A}^\mathsf{T} \right\| \)	Solució:	\( \left\|\boldsymbol{A}^\mathsf{T}\right\| = \left\|\boldsymbol{A}\right\| = 5 \)
h) \( \left\| \boldsymbol{A}^{*} \right\| \)	Solució:	\( \left\|\boldsymbol{A}^{*}\right\| = \left\|\boldsymbol{A}\right\|^{2-1} = 5 \)
i) \( \left\| \boldsymbol{A}^5 \right\| \)	Solució:	\( \left\|\boldsymbol{A}^5\right\| = \left\|\boldsymbol{A}\right\|^5 = 5^5 = 3125 \)

Exercici 6

Resol les equacións

a) \( \left\| \begin{array}{lcc} x^2&1&1 \\ 1&x&1 \\ 1&1&1 \end{array} \right\| = 0 \)	Solució:	\( x=\pm 1 \)
b) \( \left\| \begin{array}{ccc} x-1&1&1 \\ 1&x-1&1 \\ 1&1&x-1 \end{array} \right\| = 5-3x \)	Solució:	\( x=1 \)

Exercici 7

Demostra la següent igualtat, aplicant només les propietats dels determinants:

\( \left| \begin{array}{cccc} x&3&3&3 \\ 5&x&1&1 \\ 4&4&x&7 \\ 1&1&1&1 \end{array} \right| = (x-1)\cdot(x-3)\cdot(x-7) \)

Solució:

a) \( \left\| \boldsymbol{B}^2 \right\|\)	Solució:	\( \left\| \boldsymbol{B}^2 \right\| = (-5)^2 = 25 \)
b) \( \left\| 5\boldsymbol{A} \right\|\)	Solució:	\( \left\| 5\boldsymbol{A} \right\| = 5^4 \cdot \left\| \boldsymbol{A} \right\| = 625 \cdot 6 = 3750 \)
c) \( \left\| -\boldsymbol{A} \right\|\)	Solució:	\( \left\| -\boldsymbol{A} \right\| = (-1)^4 \cdot \left\| \boldsymbol{A} \right\| = 1 \cdot 6 = 6 \)
d) \( \left\| \boldsymbol{A}^\mathsf{T} \cdot \boldsymbol{B} \right\|\)	Solució:	\( \left\| \boldsymbol{A}^\mathsf{T} \cdot \boldsymbol{B} \right\| = \left\| \boldsymbol{A}^\mathsf{T} \right\| \cdot \left\| \boldsymbol{B} \right\| = \left\| \boldsymbol{A} \right\| \cdot \left\| \boldsymbol{B} \right\| = 6 \cdot (-5) = -30 \)
e) \( \left\| \boldsymbol{B}^{-1} \right\|\)	Solució:	\( \left\| \boldsymbol{B}^{-1} \right\| = \displaystyle \frac{1}{\left\| \boldsymbol{B} \right\|} = -\frac{1}{5} \)

a) \( \left\|\begin{array}{rr} 3a&3b\\c&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 3a&3b\\c&d \end{array}\right\| = 3·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 15 \)
b) \( \left\|\begin{array}{rr} -b&-a\\d&c \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} -b&-a\\d&c \end{array}\right\| = -1·\left\|\begin{array}{rr} b&a\\d&c \end{array}\right\| = \left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 5 \)
c) \( \left\|\begin{array}{rr} 4a&4b\\4c&4d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 4a&4b\\4c&4d \end{array}\right\| = 4^2·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 80 \)
d) \( \left\|\begin{array}{rr} -6a&-2b\\3c&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} -6a&-2b\\3c&d \end{array}\right\| = -2·\left\|\begin{array}{rr} 3a&b\\3c&d \end{array}\right\| = -2·3·5 = -30 \)
e) \( \left\|\begin{array}{rr} 3a-2b&b\\3c-2d&d \end{array}\right\| \)	Solució:	\( \left\|\begin{array}{rr} 3a-2b&b\\3c-2d&d \end{array}\right\| = \left\|\begin{array}{rr} 3a&b\\3c&d \end{array}\right\| = 3·\left\|\begin{array}{rr} a&b\\c&d \end{array}\right\| = 15 \)
f) \( \left\| \boldsymbol{A}^{-1} \right\| \)	Solució:	\( \displaystyle \left\|\boldsymbol{A}^{-1}\right\| = \frac{1}{\left\|\boldsymbol{A}\right\|} = \frac{1}{5} \)
g) \( \left\| \boldsymbol{A}^\mathsf{T} \right\| \)	Solució:	\( \left\|\boldsymbol{A}^\mathsf{T}\right\| = \left\|\boldsymbol{A}\right\| = 5 \)
h) \( \left\| \boldsymbol{A}^{*} \right\| \)	Solució:	\( \left\|\boldsymbol{A}^{*}\right\| = \left\|\boldsymbol{A}\right\|^{2-1} = 5 \)
i) \( \left\| \boldsymbol{A}^5 \right\| \)	Solució:	\( \left\|\boldsymbol{A}^5\right\| = \left\|\boldsymbol{A}\right\|^5 = 5^5 = 3125 \)