Soalan 1:
$$ \text { Diberi empat matriks } P=\left[\begin{array}{cc} 3 & 6 \\ -1 & 2 \end{array}\right], Q=\left[\begin{array}{l} 7 \\ 9 \end{array}\right], R=\left[\begin{array}{lll} 4 & 8 & 5 \end{array}\right] \text { dan } S=\left[\begin{array}{ccc} 0 & -6 & 1 \\ 3 & 11 & -2 \end{array}\right] \text {. } $$
Tentukan sama ada pendaraban matriks berikut boleh dilakukan atau tidak. Jika ya, nyatakan peringkat hasil darab pasangan matriks itu.
(a) PQ
(b) QR
(c) RS
(d) SP
(e) PS
(f) QP
Penyelesaian:
(a)
(b)
(c)
(d)
(e)
(f)
$$ \text { Diberi empat matriks } P=\left[\begin{array}{cc} 3 & 6 \\ -1 & 2 \end{array}\right], Q=\left[\begin{array}{l} 7 \\ 9 \end{array}\right], R=\left[\begin{array}{lll} 4 & 8 & 5 \end{array}\right] \text { dan } S=\left[\begin{array}{ccc} 0 & -6 & 1 \\ 3 & 11 & -2 \end{array}\right] \text {. } $$
Tentukan sama ada pendaraban matriks berikut boleh dilakukan atau tidak. Jika ya, nyatakan peringkat hasil darab pasangan matriks itu.
(a) PQ
(b) QR
(c) RS
(d) SP
(e) PS
(f) QP
Penyelesaian:
(a)
(b)(c)
(d)(e)
(f)Soalan 2:
$$ \begin{aligned} & \text { Diberi empat matriks, } T=\left[\begin{array}{ccc} 1 & 3 & 4 \\ -2 & 2 & -1 \end{array}\right], U=\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right], V=\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \text { dan } W=\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] . \\ & \text { Hitung } \end{aligned} $$
(a) TU
(b) UW
(c) UV
(d) WV
(e) W2
(f) W3
Penyelesaian:
(a)
$$ \begin{aligned} T U & =\left[\begin{array}{ccc} 1 & 3 & 4 \\ -2 & 2 & -1 \end{array}\right]\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 1(0)+3(-3)+4(1) & 1(-4)+3(5)+4(2) \\ (-2) 0+2(-3)+(-1) 1 & (-2)(-4)+2(5)+(-1) 2 \end{array}\right] \\ & =\left[\begin{array}{cc} -5 & 19 \\ -7 & 16 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} U W & =\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 0(2)+(-4) 3 & 0(1)+(-4)(-4) \\ (-3) 2+5(3) & (-3) 1+5(-4) \\ 1(2)+2(3) & 1(1)+2(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} -12 & 16 \\ 9 & -23 \\ 8 & -7 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} U V & =\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right]\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \\ & =\left[\begin{array}{c} 0(-6)+(-4) 2 \\ (-3)(-6)+5(2) \\ 1(-6)+2(2) \end{array}\right] \\ & =\left[\begin{array}{c} -8 \\ 28 \\ -2 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} W V & =\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right]\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \\ & =\left[\begin{array}{c} 2(-6)+1(2) \\ 3(-6)+(-4) 2 \end{array}\right] \\ & =\left[\begin{array}{l} -10 \\ -26 \end{array}\right] \end{aligned} $$
(e)
$$ \begin{aligned} W^2 & =W W \\ & =\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 2(2)+1(3) & 2(1)+1(-4) \\ 3(2)+(-4) 3 & 3(1)+(-4)(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & -2 \\ -6 & 19 \end{array}\right] \end{aligned} $$
(f)
$$ \begin{aligned} W^3 & =W^2 W \\ & =\left[\begin{array}{cc} 7 & -2 \\ -6 & 19 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(2)+(-2) 3 & 7(1)+(-2)(-4) \\ (-6) 2+19(3) & (-6) 1+19(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} 8 & 15 \\ 45 & -82 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} & \text { Diberi empat matriks, } T=\left[\begin{array}{ccc} 1 & 3 & 4 \\ -2 & 2 & -1 \end{array}\right], U=\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right], V=\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \text { dan } W=\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] . \\ & \text { Hitung } \end{aligned} $$
(a) TU
(b) UW
(c) UV
(d) WV
(e) W2
(f) W3
Penyelesaian:
(a)
$$ \begin{aligned} T U & =\left[\begin{array}{ccc} 1 & 3 & 4 \\ -2 & 2 & -1 \end{array}\right]\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right] \\ & =\left[\begin{array}{cc} 1(0)+3(-3)+4(1) & 1(-4)+3(5)+4(2) \\ (-2) 0+2(-3)+(-1) 1 & (-2)(-4)+2(5)+(-1) 2 \end{array}\right] \\ & =\left[\begin{array}{cc} -5 & 19 \\ -7 & 16 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} U W & =\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 0(2)+(-4) 3 & 0(1)+(-4)(-4) \\ (-3) 2+5(3) & (-3) 1+5(-4) \\ 1(2)+2(3) & 1(1)+2(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} -12 & 16 \\ 9 & -23 \\ 8 & -7 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} U V & =\left[\begin{array}{cc} 0 & -4 \\ -3 & 5 \\ 1 & 2 \end{array}\right]\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \\ & =\left[\begin{array}{c} 0(-6)+(-4) 2 \\ (-3)(-6)+5(2) \\ 1(-6)+2(2) \end{array}\right] \\ & =\left[\begin{array}{c} -8 \\ 28 \\ -2 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} W V & =\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right]\left[\begin{array}{c} -6 \\ 2 \end{array}\right] \\ & =\left[\begin{array}{c} 2(-6)+1(2) \\ 3(-6)+(-4) 2 \end{array}\right] \\ & =\left[\begin{array}{l} -10 \\ -26 \end{array}\right] \end{aligned} $$
(e)
$$ \begin{aligned} W^2 & =W W \\ & =\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 2(2)+1(3) & 2(1)+1(-4) \\ 3(2)+(-4) 3 & 3(1)+(-4)(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} 7 & -2 \\ -6 & 19 \end{array}\right] \end{aligned} $$
(f)
$$ \begin{aligned} W^3 & =W^2 W \\ & =\left[\begin{array}{cc} 7 & -2 \\ -6 & 19 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ 3 & -4 \end{array}\right] \\ & =\left[\begin{array}{cc} 7(2)+(-2) 3 & 7(1)+(-2)(-4) \\ (-6) 2+19(3) & (-6) 1+19(-4) \end{array}\right] \\ & =\left[\begin{array}{cc} 8 & 15 \\ 45 & -82 \end{array}\right] \end{aligned} $$