Soalan 1:
Tentukan hasil darab bagi setiap matriks berikut.
$$ \text { (a) } 3\left[\begin{array}{c} -7 \\ 2 \end{array}\right] $$
$$ \text { (b) } 0.6\left[\begin{array}{ll} 11 & 5 \end{array}\right] $$
$$ \text { (c) } \frac{1}{4}\left[\begin{array}{cc} 12 & -20 \\ -6 & 16 \\ 9 & 1 \end{array}\right] $$
$$ \text { (d) }-2\left[\begin{array}{cc} 0.4 & 8 \\ -9 & 2.5 \end{array}\right] $$
$$ \text { (e) } \quad 1.2\left[\begin{array}{ccc} 10 & -1 & 11 \\ 3 & 7 & -5 \end{array}\right] $$
$$ \text { (f) } \quad-\frac{1}{20}\left[\begin{array}{l} 100 \\ -90 \\ -20 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} 3\left[\begin{array}{c} -7 \\ 2 \end{array}\right] & =\left[\begin{array}{c} 3(-7) \\ 3(2) \end{array}\right] \\ & =\left[\begin{array}{c} -21 \\ 6 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} 0.6\left[\begin{array}{ll} 11 & 5 \end{array}\right] & =\left[\begin{array}{ll} 0.6(11) & 0.6(5) \end{array}\right] \\ & =\left[\begin{array}{ll} 6.6 & 3 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{1}{4}\left[\begin{array}{cc} 12 & -20 \\ -6 & 16 \\ 9 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{4}(12) & \frac{1}{4}(-20) \\ \frac{1}{4}(-6) & \frac{1}{4}(16) \\ \frac{1}{4}(9) & \frac{1}{4}(1) \end{array}\right] \\ & =\left[\begin{array}{cc} 3 & -5 \\ -\frac{3}{2} & 4 \\ \frac{9}{4} & \frac{1}{4} \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & -2\left[\begin{array}{cc} 0.4 & 8 \\ -9 & 2.5 \end{array}\right] \\ & =\left[\begin{array}{cc} -2(0.4) & -2(8) \\ -2(-9) & -2(2.5) \end{array}\right] \\ & =\left[\begin{array}{cc} -0.8 & -16 \\ 18 & -5 \end{array}\right] \end{aligned} $$
(e)
$$ \begin{aligned} & 1.2\left[\begin{array}{ccc} 10 & -1 & 11 \\ 3 & 7 & -5 \end{array}\right] \\ & =\left[\begin{array}{lll} 1.2(10) & 1.2(-1) & 1.2(11) \\ 1.2(3) & 1.2(7) & 1.2(-5) \end{array}\right] \\ & =\left[\begin{array}{lrr} 12 & -1.2 & 13.2 \\ 3.6 & 8.4 & -6 \end{array}\right] \end{aligned} $$
(f)
$$ \begin{aligned} & -\frac{1}{20}\left[\begin{array}{c} 100 \\ -90 \\ -20 \end{array}\right] \\ & =\left[\begin{array}{c} -\frac{1}{20}(100) \\ -\frac{1}{20}(-90) \\ -\frac{1}{20}(-20) \end{array}\right] \\ & =\left[\begin{array}{c} -5 \\ \frac{9}{2} \\ 1 \end{array}\right] \end{aligned} $$
Tentukan hasil darab bagi setiap matriks berikut.
$$ \text { (a) } 3\left[\begin{array}{c} -7 \\ 2 \end{array}\right] $$
$$ \text { (b) } 0.6\left[\begin{array}{ll} 11 & 5 \end{array}\right] $$
$$ \text { (c) } \frac{1}{4}\left[\begin{array}{cc} 12 & -20 \\ -6 & 16 \\ 9 & 1 \end{array}\right] $$
$$ \text { (d) }-2\left[\begin{array}{cc} 0.4 & 8 \\ -9 & 2.5 \end{array}\right] $$
$$ \text { (e) } \quad 1.2\left[\begin{array}{ccc} 10 & -1 & 11 \\ 3 & 7 & -5 \end{array}\right] $$
$$ \text { (f) } \quad-\frac{1}{20}\left[\begin{array}{l} 100 \\ -90 \\ -20 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} 3\left[\begin{array}{c} -7 \\ 2 \end{array}\right] & =\left[\begin{array}{c} 3(-7) \\ 3(2) \end{array}\right] \\ & =\left[\begin{array}{c} -21 \\ 6 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} 0.6\left[\begin{array}{ll} 11 & 5 \end{array}\right] & =\left[\begin{array}{ll} 0.6(11) & 0.6(5) \end{array}\right] \\ & =\left[\begin{array}{ll} 6.6 & 3 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{1}{4}\left[\begin{array}{cc} 12 & -20 \\ -6 & 16 \\ 9 & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} \frac{1}{4}(12) & \frac{1}{4}(-20) \\ \frac{1}{4}(-6) & \frac{1}{4}(16) \\ \frac{1}{4}(9) & \frac{1}{4}(1) \end{array}\right] \\ & =\left[\begin{array}{cc} 3 & -5 \\ -\frac{3}{2} & 4 \\ \frac{9}{4} & \frac{1}{4} \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & -2\left[\begin{array}{cc} 0.4 & 8 \\ -9 & 2.5 \end{array}\right] \\ & =\left[\begin{array}{cc} -2(0.4) & -2(8) \\ -2(-9) & -2(2.5) \end{array}\right] \\ & =\left[\begin{array}{cc} -0.8 & -16 \\ 18 & -5 \end{array}\right] \end{aligned} $$
(e)
$$ \begin{aligned} & 1.2\left[\begin{array}{ccc} 10 & -1 & 11 \\ 3 & 7 & -5 \end{array}\right] \\ & =\left[\begin{array}{lll} 1.2(10) & 1.2(-1) & 1.2(11) \\ 1.2(3) & 1.2(7) & 1.2(-5) \end{array}\right] \\ & =\left[\begin{array}{lrr} 12 & -1.2 & 13.2 \\ 3.6 & 8.4 & -6 \end{array}\right] \end{aligned} $$
(f)
$$ \begin{aligned} & -\frac{1}{20}\left[\begin{array}{c} 100 \\ -90 \\ -20 \end{array}\right] \\ & =\left[\begin{array}{c} -\frac{1}{20}(100) \\ -\frac{1}{20}(-90) \\ -\frac{1}{20}(-20) \end{array}\right] \\ & =\left[\begin{array}{c} -5 \\ \frac{9}{2} \\ 1 \end{array}\right] \end{aligned} $$
Soalan 2:
Selesaikan setiap operasi yang berikut.
$$ \text { (a) } 5\left[\begin{array}{cc} 1 & 2 \\ 3 & -1 \\ 0 & -4 \end{array}\right]+\left[\begin{array}{cc} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\frac{1}{2}\left[\begin{array}{cc} 10 & 2 \\ 9 & -4 \\ -6 & 14 \end{array}\right] $$
$$ \text { (b) } \quad 6\left[\begin{array}{c} -1 \\ 4 \end{array}\right]-0.5\left[\begin{array}{c} 8 \\ 14 \end{array}\right]-2\left[\begin{array}{c} 1 \\ -3 \end{array}\right] $$
$$ \text { (c) } 7\left[\begin{array}{lll} 3 & -2 & 1 \end{array}\right]-\frac{1}{3}\left[\begin{array}{lll} 21 & 6 & -9 \end{array}\right] $$
$$ \text { (d) } \quad 0.2\left[\begin{array}{cc} 10 & -25 \\ -6 & 8 \end{array}\right]+\frac{1}{5}\left[\begin{array}{cc} 15 & 20 \\ -5 & 2.5 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} & 5\left[\begin{array}{ll} 1 & 2 \\ 3 & -1 \\ 0 & -4 \end{array}\right]+\left[\begin{array}{ll} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\frac{1}{2}\left[\begin{array}{cc} 10 & 2 \\ 9 & -4 \\ -6 & 14 \end{array}\right] \\ & =\left[\begin{array}{cc} 5 & 10 \\ 15 & -5 \\ 0 & -20 \end{array}\right]+\left[\begin{array}{cc} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\left[\begin{array}{cc} 5 & 1 \\ 9 & -2 \\ \frac{9}{2} & 7 \end{array}\right] \\ & =\left[\begin{array}{cc} 5+7-5 & 10+5-1 \\ 15+6-\frac{9}{2} & -5+1+2 \\ 0-1+3 & -20+8-7 \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & 14 \\ \frac{33}{2} & -2 \\ 2 & -19 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} 6\left[\begin{array}{c} -1 \\ 4 \end{array}\right]-0.5\left[\begin{array}{c} 8 \\ 14 \end{array}\right]-2\left[\begin{array}{c} 1 \\ -3 \end{array}\right] & =\left[\begin{array}{c} -6 \\ 24 \end{array}\right]-\left[\begin{array}{l} 4 \\ 7 \end{array}\right]-\left[\begin{array}{c} 2 \\ -6 \end{array}\right] \\ & =\left[\begin{array}{c} -6-4-2 \\ 24-7+6 \end{array}\right] \\ & =\left[\begin{array}{c} -12 \\ 23 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & 7\left[\begin{array}{lll} 3 & -2 & 1 \end{array}\right]-\frac{1}{3}\left[\begin{array}{lll} 21 & 6 & -9 \end{array}\right] \\ & =\left[\begin{array}{lll} 21 & -14 & 7 \end{array}\right]-\left[\begin{array}{lll} 7 & 2 & -3 \end{array}\right] \\ & =\left[\begin{array}{lll} 21-7 & -14-2 & 7-(-3) \end{array}\right] \\ & =\left[\begin{array}{lll} 14 & -16 & 10 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & 0.2\left[\begin{array}{cc} 10 & -25 \\ -6 & 8 \end{array}\right]+\frac{1}{5}\left[\begin{array}{cc} 15 & 20 \\ -5 & 2.5 \end{array}\right] \\ & \quad=\left[\begin{array}{cc} 2 & -5 \\ -1.2 & 1.6 \end{array}\right]+\left[\begin{array}{cc} 3 & 4 \\ -1 & 0.5 \end{array}\right] \\ & \quad=\left[\begin{array}{cc} 5 & -1 \\ -2.2 & 2.1 \end{array}\right] \end{aligned} $$
Selesaikan setiap operasi yang berikut.
$$ \text { (a) } 5\left[\begin{array}{cc} 1 & 2 \\ 3 & -1 \\ 0 & -4 \end{array}\right]+\left[\begin{array}{cc} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\frac{1}{2}\left[\begin{array}{cc} 10 & 2 \\ 9 & -4 \\ -6 & 14 \end{array}\right] $$
$$ \text { (b) } \quad 6\left[\begin{array}{c} -1 \\ 4 \end{array}\right]-0.5\left[\begin{array}{c} 8 \\ 14 \end{array}\right]-2\left[\begin{array}{c} 1 \\ -3 \end{array}\right] $$
$$ \text { (c) } 7\left[\begin{array}{lll} 3 & -2 & 1 \end{array}\right]-\frac{1}{3}\left[\begin{array}{lll} 21 & 6 & -9 \end{array}\right] $$
$$ \text { (d) } \quad 0.2\left[\begin{array}{cc} 10 & -25 \\ -6 & 8 \end{array}\right]+\frac{1}{5}\left[\begin{array}{cc} 15 & 20 \\ -5 & 2.5 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} & 5\left[\begin{array}{ll} 1 & 2 \\ 3 & -1 \\ 0 & -4 \end{array}\right]+\left[\begin{array}{ll} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\frac{1}{2}\left[\begin{array}{cc} 10 & 2 \\ 9 & -4 \\ -6 & 14 \end{array}\right] \\ & =\left[\begin{array}{cc} 5 & 10 \\ 15 & -5 \\ 0 & -20 \end{array}\right]+\left[\begin{array}{cc} 7 & 5 \\ 6 & 1 \\ -1 & 8 \end{array}\right]-\left[\begin{array}{cc} 5 & 1 \\ 9 & -2 \\ \frac{9}{2} & 7 \end{array}\right] \\ & =\left[\begin{array}{cc} 5+7-5 & 10+5-1 \\ 15+6-\frac{9}{2} & -5+1+2 \\ 0-1+3 & -20+8-7 \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & 14 \\ \frac{33}{2} & -2 \\ 2 & -19 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} 6\left[\begin{array}{c} -1 \\ 4 \end{array}\right]-0.5\left[\begin{array}{c} 8 \\ 14 \end{array}\right]-2\left[\begin{array}{c} 1 \\ -3 \end{array}\right] & =\left[\begin{array}{c} -6 \\ 24 \end{array}\right]-\left[\begin{array}{l} 4 \\ 7 \end{array}\right]-\left[\begin{array}{c} 2 \\ -6 \end{array}\right] \\ & =\left[\begin{array}{c} -6-4-2 \\ 24-7+6 \end{array}\right] \\ & =\left[\begin{array}{c} -12 \\ 23 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & 7\left[\begin{array}{lll} 3 & -2 & 1 \end{array}\right]-\frac{1}{3}\left[\begin{array}{lll} 21 & 6 & -9 \end{array}\right] \\ & =\left[\begin{array}{lll} 21 & -14 & 7 \end{array}\right]-\left[\begin{array}{lll} 7 & 2 & -3 \end{array}\right] \\ & =\left[\begin{array}{lll} 21-7 & -14-2 & 7-(-3) \end{array}\right] \\ & =\left[\begin{array}{lll} 14 & -16 & 10 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & 0.2\left[\begin{array}{cc} 10 & -25 \\ -6 & 8 \end{array}\right]+\frac{1}{5}\left[\begin{array}{cc} 15 & 20 \\ -5 & 2.5 \end{array}\right] \\ & \quad=\left[\begin{array}{cc} 2 & -5 \\ -1.2 & 1.6 \end{array}\right]+\left[\begin{array}{cc} 3 & 4 \\ -1 & 0.5 \end{array}\right] \\ & \quad=\left[\begin{array}{cc} 5 & -1 \\ -2.2 & 2.1 \end{array}\right] \end{aligned} $$
Soalan 3:
$$ \text { Diberi matriks } E=\left[\begin{array}{cc} 9 & 6 \\ 2 & 11 \end{array}\right] \text {, matriks } F=\left[\begin{array}{cc} -7 & 22 \\ 3 & 4 \end{array}\right] \text { dan matriks } G=\left[\begin{array}{cc} -1 & 10 \\ -8 & 5 \end{array}\right] \text {, } $$
$$ \text { tunjukkan }(E+F)+G=E+(F+G) \text {. } $$
Penyelesaian:
$$ \begin{aligned} (E+F)+G & =\left(\left[\begin{array}{rr} 9 & 6 \\ 2 & 11 \end{array}\right]+\left[\begin{array}{rr} -7 & 22 \\ 3 & 4 \end{array}\right]\right)+\left[\begin{array}{rr} -1 & 10 \\ -8 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 2 & 28 \\ 5 & 15 \end{array}\right]+\left[\begin{array}{cc} -1 & 10 \\ -8 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & 38 \\ -3 & 20 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} E+(F+G) & =\left[\begin{array}{rr} 9 & 6 \\ 2 & 11 \end{array}\right]+\left(\left[\begin{array}{cc} -7 & 22 \\ 3 & 4 \end{array}\right]+\left[\begin{array}{rr} -1 & 10 \\ -8 & 5 \end{array}\right]\right) \\ & =\left[\begin{array}{cc} 9 & 6 \\ 2 & 11 \end{array}\right]+\left[\begin{array}{rr} -8 & 32 \\ -5 & 9 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & 38 \\ -3 & 20 \end{array}\right] \end{aligned} $$
$$ \text { Maka, }(E+F)+G=E+(F+G) \quad \text { (Tertunjuk) } $$
$$ \text { Diberi matriks } E=\left[\begin{array}{cc} 9 & 6 \\ 2 & 11 \end{array}\right] \text {, matriks } F=\left[\begin{array}{cc} -7 & 22 \\ 3 & 4 \end{array}\right] \text { dan matriks } G=\left[\begin{array}{cc} -1 & 10 \\ -8 & 5 \end{array}\right] \text {, } $$
$$ \text { tunjukkan }(E+F)+G=E+(F+G) \text {. } $$
Penyelesaian:
$$ \begin{aligned} (E+F)+G & =\left(\left[\begin{array}{rr} 9 & 6 \\ 2 & 11 \end{array}\right]+\left[\begin{array}{rr} -7 & 22 \\ 3 & 4 \end{array}\right]\right)+\left[\begin{array}{rr} -1 & 10 \\ -8 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 2 & 28 \\ 5 & 15 \end{array}\right]+\left[\begin{array}{cc} -1 & 10 \\ -8 & 5 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & 38 \\ -3 & 20 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} E+(F+G) & =\left[\begin{array}{rr} 9 & 6 \\ 2 & 11 \end{array}\right]+\left(\left[\begin{array}{cc} -7 & 22 \\ 3 & 4 \end{array}\right]+\left[\begin{array}{rr} -1 & 10 \\ -8 & 5 \end{array}\right]\right) \\ & =\left[\begin{array}{cc} 9 & 6 \\ 2 & 11 \end{array}\right]+\left[\begin{array}{rr} -8 & 32 \\ -5 & 9 \end{array}\right] \\ & =\left[\begin{array}{cc} 1 & 38 \\ -3 & 20 \end{array}\right] \end{aligned} $$
$$ \text { Maka, }(E+F)+G=E+(F+G) \quad \text { (Tertunjuk) } $$