Soalan 1:
Antara matriks berikut, yang manakah matriks identiti? Jika bukan, berikan sebab anda.
Penyelesaian:
(a)
[0 1 ]
Bukan. Ini bukan matriks segi empat sama.
(b)
Bukan. Unsur di pepenjuru utama bukan 1.
(c)
Bukan. Unsur di pepenjuru utama bukan 1.
(d)
Ya, matriks identiti.
(e)
Ya, matriks identiti.
(f)
Bukan. Unsur di pepenjuru utama bukan 1.
Antara matriks berikut, yang manakah matriks identiti? Jika bukan, berikan sebab anda.
Penyelesaian:
(a)
[0 1 ]
Bukan. Ini bukan matriks segi empat sama.
(b)
Bukan. Unsur di pepenjuru utama bukan 1.
(c)
Bukan. Unsur di pepenjuru utama bukan 1.
(d)
Ya, matriks identiti.
(e)
Ya, matriks identiti.
(f)
Bukan. Unsur di pepenjuru utama bukan 1.
Soalan 2:
$$ \text { Diberi matriks } C=\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \text { dan matriks } D=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \text {. } $$
Tunjukkan matriks D ialah matriks identiti.
Penyelesaian:
$$ \begin{aligned} C D & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} -1(1)+3(0) & (-1) 0+3(1) \\ 2(1)+5(0) & 2(0)+5(1) \end{array}\right] \\ & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} D C & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1(-1)+0(2) & 1(3)+0(5) \\ 0(-1)+1(2) & 0(3)+1(5) \end{array}\right] \\ & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \end{aligned} $$
$$ C D=D C=C \text {. Maka, } D \text { ialah matriks identiti. } $$
$$ \text { Diberi matriks } C=\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \text { dan matriks } D=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \text {. } $$
Tunjukkan matriks D ialah matriks identiti.
Penyelesaian:
$$ \begin{aligned} C D & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} -1(1)+3(0) & (-1) 0+3(1) \\ 2(1)+5(0) & 2(0)+5(1) \end{array}\right] \\ & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} D C & =\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1(-1)+0(2) & 1(3)+0(5) \\ 0(-1)+1(2) & 0(3)+1(5) \end{array}\right] \\ & =\left[\begin{array}{cc} -1 & 3 \\ 2 & 5 \end{array}\right] \end{aligned} $$
$$ C D=D C=C \text {. Maka, } D \text { ialah matriks identiti. } $$
Soalan 3:
$$ \text { Diberi matriks } S=\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right] \text { dan matriks } T=\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \text {. Hitung } $$
(a) SI + TI
(b) (IS)T
(c) 4IT – I2
(d) (S – I)I
Penyelesaian:
(a)
$$ \begin{aligned} & S I+T I \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]+\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]+\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7+3 & 2+1 \\ 6+(-5) & 3+4 \end{array}\right] \\ & =\left[\begin{array}{cc} 10 & 3 \\ 1 & 7 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} & (I S) T \\ & =\left(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\right)\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(3)+2(-5) & 7(1)+2(4) \\ 6(3)+3(-5) & 6(1)+3(4) \end{array}\right] \\ & =\left[\begin{array}{cc} 11 & 15 \\ 3 & 18 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} 4 I T-I^2 & =4 T-I \\ & =4\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} 12 & 4 \\ -20 & 16 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} 11 & 4 \\ -20 & 15 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} (S-I) I & =S-I \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{ll} 6 & 2 \\ 6 & 2 \end{array}\right] \end{aligned} $$
$$ \text { Diberi matriks } S=\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right] \text { dan matriks } T=\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \text {. Hitung } $$
(a) SI + TI
(b) (IS)T
(c) 4IT – I2
(d) (S – I)I
Penyelesaian:
(a)
$$ \begin{aligned} & S I+T I \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]+\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]+\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7+3 & 2+1 \\ 6+(-5) & 3+4 \end{array}\right] \\ & =\left[\begin{array}{cc} 10 & 3 \\ 1 & 7 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} & (I S) T \\ & =\left(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\right)\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(3)+2(-5) & 7(1)+2(4) \\ 6(3)+3(-5) & 6(1)+3(4) \end{array}\right] \\ & =\left[\begin{array}{cc} 11 & 15 \\ 3 & 18 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} 4 I T-I^2 & =4 T-I \\ & =4\left[\begin{array}{cc} 3 & 1 \\ -5 & 4 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} 12 & 4 \\ -20 & 16 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} 11 & 4 \\ -20 & 15 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} (S-I) I & =S-I \\ & =\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & =\left[\begin{array}{ll} 6 & 2 \\ 6 & 2 \end{array}\right] \end{aligned} $$