Soalan 1:
Tentukan sama ada penambahan dan penolakan boleh dilaksanakan pada setiap pasangan matriks berikut.
$$ \text { (a) }\left[\begin{array}{cc} -5 & 9 \\ 1 & 0 \end{array}\right] \text { dan }\left[\begin{array}{ll} 3 & 7 \\ 6 & 1 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{ccc} 13 & -1 & 11 \\ -2 & 8 & 4 \end{array}\right] \operatorname{dan}\left[\begin{array}{ccc} 4 & -16 & 7 \\ 1 & 5 & 0 \\ 3 & 2 & 8 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{c} 10 \\ -3 \end{array}\right] \text { dan }\left[\begin{array}{ll} 12 & -7 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} 2 & -9 \end{array}\right] \text { dan }\left[\begin{array}{ll} 1 & 6 \end{array}\right] $$
Penyelesaian:
(a) Boleh kerana peringkat kedua-dua matriks adalah sama.
(b) Tidak boleh kerana peringkat kedua-dua matriks adalah tidak sama.
(c) Tidak boleh kerana peringkat kedua-dua matriks adalah tidak sama.
(d) Boleh kerana peringkat kedua-dua matriks adalah sama.
Tentukan sama ada penambahan dan penolakan boleh dilaksanakan pada setiap pasangan matriks berikut.
$$ \text { (a) }\left[\begin{array}{cc} -5 & 9 \\ 1 & 0 \end{array}\right] \text { dan }\left[\begin{array}{ll} 3 & 7 \\ 6 & 1 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{ccc} 13 & -1 & 11 \\ -2 & 8 & 4 \end{array}\right] \operatorname{dan}\left[\begin{array}{ccc} 4 & -16 & 7 \\ 1 & 5 & 0 \\ 3 & 2 & 8 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{c} 10 \\ -3 \end{array}\right] \text { dan }\left[\begin{array}{ll} 12 & -7 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} 2 & -9 \end{array}\right] \text { dan }\left[\begin{array}{ll} 1 & 6 \end{array}\right] $$
Penyelesaian:
(a) Boleh kerana peringkat kedua-dua matriks adalah sama.
(b) Tidak boleh kerana peringkat kedua-dua matriks adalah tidak sama.
(c) Tidak boleh kerana peringkat kedua-dua matriks adalah tidak sama.
(d) Boleh kerana peringkat kedua-dua matriks adalah sama.
Soalan 2:
$$ \text { Diberi matriks } P=\left[\begin{array}{cc} 12 & 1 \\ -3 & 4 \end{array}\right], Q=\left[\begin{array}{cc} 8 & -2 \\ 0 & 5 \end{array}\right] \text { dan } R=\left[\begin{array}{cc} 6 & 3 \\ 7 & -1 \end{array}\right] \text {, hitung } $$
(a) P – Q + R
(b) P + Q – R
Penyelesaian:
(a)
$$ \begin{aligned} P-Q+R & =\left[\begin{array}{rr} 12 & 1 \\ -3 & 4 \end{array}\right]-\left[\begin{array}{rr} 8 & -2 \\ 0 & 5 \end{array}\right]+\left[\begin{array}{rr} 6 & 3 \\ 7 & -1 \end{array}\right] \\ & =\left[\begin{array}{rr} 12-8+6 & 1-(-2)+3 \\ -3-0+7 & 4-5+(-1) \end{array}\right] \\ & =\left[\begin{array}{cr} 10 & 6 \\ 4 & -2 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} P+Q-R & =\left[\begin{array}{ll} 12 & 1 \\ -3 & 4 \end{array}\right]+\left[\begin{array}{rr} 8 & -2 \\ 0 & 5 \end{array}\right]-\left[\begin{array}{rr} 6 & 3 \\ 7 & -1 \end{array}\right] \\ & =\left[\begin{array}{ll} 12+8-6 & 1+(-2)-3 \\ -3+0-7 & 4+5-(-1) \end{array}\right] \\ & =\left[\begin{array}{ll} 14 & -4 \\ -10 & 10 \end{array}\right] \end{aligned} $$
$$ \text { Diberi matriks } P=\left[\begin{array}{cc} 12 & 1 \\ -3 & 4 \end{array}\right], Q=\left[\begin{array}{cc} 8 & -2 \\ 0 & 5 \end{array}\right] \text { dan } R=\left[\begin{array}{cc} 6 & 3 \\ 7 & -1 \end{array}\right] \text {, hitung } $$
(a) P – Q + R
(b) P + Q – R
Penyelesaian:
(a)
$$ \begin{aligned} P-Q+R & =\left[\begin{array}{rr} 12 & 1 \\ -3 & 4 \end{array}\right]-\left[\begin{array}{rr} 8 & -2 \\ 0 & 5 \end{array}\right]+\left[\begin{array}{rr} 6 & 3 \\ 7 & -1 \end{array}\right] \\ & =\left[\begin{array}{rr} 12-8+6 & 1-(-2)+3 \\ -3-0+7 & 4-5+(-1) \end{array}\right] \\ & =\left[\begin{array}{cr} 10 & 6 \\ 4 & -2 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} P+Q-R & =\left[\begin{array}{ll} 12 & 1 \\ -3 & 4 \end{array}\right]+\left[\begin{array}{rr} 8 & -2 \\ 0 & 5 \end{array}\right]-\left[\begin{array}{rr} 6 & 3 \\ 7 & -1 \end{array}\right] \\ & =\left[\begin{array}{ll} 12+8-6 & 1+(-2)-3 \\ -3+0-7 & 4+5-(-1) \end{array}\right] \\ & =\left[\begin{array}{ll} 14 & -4 \\ -10 & 10 \end{array}\right] \end{aligned} $$
Soalan 3:
Selesaikan setiap yang berikut.
$$ \text { (a) }\left[\begin{array}{ccc} 12 & 10 & 1 \\ -4 & 0 & -7 \end{array}\right]+\left[\begin{array}{ccc} 1 & -1 & 9 \\ 2 & 8 & 3 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{rr} 18 & -3 \\ -7 & 15 \end{array}\right]-\left[\begin{array}{rr} 11 & 5 \\ -1 & 2 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{c} -1 \\ 6 \end{array}\right]+\left[\begin{array}{l} 4 \\ 9 \end{array}\right]-\left[\begin{array}{c} 19 \\ -3 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} 2 & 8 \\ 1 & 5 \end{array}\right]+\left[\begin{array}{cc} 14 & 6 \\ -1 & 15 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} & {\left[\begin{array}{ccc} 12 & 10 & 1 \\ -4 & 0 & -7 \end{array}\right]+\left[\begin{array}{ccc} 1 & -1 & 9 \\ 2 & 8 & 3 \end{array}\right]} \\ & =\left[\begin{array}{ccc} 12+1 & 10+(-1) & 1+9 \\ -4+2 & 0+8 & -7+3 \end{array}\right] \\ & =\left[\begin{array}{ccc} 13 & 9 & 10 \\ -2 & 8 & -4 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} & {\left[\begin{array}{cc} 18 & -3 \\ -7 & 15 \end{array}\right]-\left[\begin{array}{ll} 11 & 5 \\ -1 & 2 \end{array}\right]} \\ & =\left[\begin{array}{cc} 18-11 & -3-5 \\ -7-(-1) & 15-2 \end{array}\right] \\ & =\left[\begin{array}{cr} 7 & -8 \\ -6 & 13 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & {\left[\begin{array}{c} -1 \\ 6 \end{array}\right]+\left[\begin{array}{l} 4 \\ 9 \end{array}\right]-\left[\begin{array}{l} 19 \\ -3 \end{array}\right]} \\ & =\left[\begin{array}{c} -1+4-19 \\ 6+9-(-3) \end{array}\right] \\ & =\left[\begin{array}{c} -16 \\ 18 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & {\left[\begin{array}{cc} 2 & 8 \\ 1 & 5 \end{array}\right]+\left[\begin{array}{cc} 14 & 6 \\ -1 & 15 \end{array}\right]} \\ & =\left[\begin{array}{cc} 2+14 & 8+6 \\ 1+(-1) & 5+15 \end{array}\right] \\ & =\left[\begin{array}{cc} 16 & 14 \\ 0 & 20 \end{array}\right] \end{aligned} $$
Selesaikan setiap yang berikut.
$$ \text { (a) }\left[\begin{array}{ccc} 12 & 10 & 1 \\ -4 & 0 & -7 \end{array}\right]+\left[\begin{array}{ccc} 1 & -1 & 9 \\ 2 & 8 & 3 \end{array}\right] $$
$$ \text { (b) }\left[\begin{array}{rr} 18 & -3 \\ -7 & 15 \end{array}\right]-\left[\begin{array}{rr} 11 & 5 \\ -1 & 2 \end{array}\right] $$
$$ \text { (c) }\left[\begin{array}{c} -1 \\ 6 \end{array}\right]+\left[\begin{array}{l} 4 \\ 9 \end{array}\right]-\left[\begin{array}{c} 19 \\ -3 \end{array}\right] $$
$$ \text { (d) }\left[\begin{array}{ll} 2 & 8 \\ 1 & 5 \end{array}\right]+\left[\begin{array}{cc} 14 & 6 \\ -1 & 15 \end{array}\right] $$
Penyelesaian:
(a)
$$ \begin{aligned} & {\left[\begin{array}{ccc} 12 & 10 & 1 \\ -4 & 0 & -7 \end{array}\right]+\left[\begin{array}{ccc} 1 & -1 & 9 \\ 2 & 8 & 3 \end{array}\right]} \\ & =\left[\begin{array}{ccc} 12+1 & 10+(-1) & 1+9 \\ -4+2 & 0+8 & -7+3 \end{array}\right] \\ & =\left[\begin{array}{ccc} 13 & 9 & 10 \\ -2 & 8 & -4 \end{array}\right] \end{aligned} $$
(b)
$$ \begin{aligned} & {\left[\begin{array}{cc} 18 & -3 \\ -7 & 15 \end{array}\right]-\left[\begin{array}{ll} 11 & 5 \\ -1 & 2 \end{array}\right]} \\ & =\left[\begin{array}{cc} 18-11 & -3-5 \\ -7-(-1) & 15-2 \end{array}\right] \\ & =\left[\begin{array}{cr} 7 & -8 \\ -6 & 13 \end{array}\right] \end{aligned} $$
(c)
$$ \begin{aligned} & {\left[\begin{array}{c} -1 \\ 6 \end{array}\right]+\left[\begin{array}{l} 4 \\ 9 \end{array}\right]-\left[\begin{array}{l} 19 \\ -3 \end{array}\right]} \\ & =\left[\begin{array}{c} -1+4-19 \\ 6+9-(-3) \end{array}\right] \\ & =\left[\begin{array}{c} -16 \\ 18 \end{array}\right] \end{aligned} $$
(d)
$$ \begin{aligned} & {\left[\begin{array}{cc} 2 & 8 \\ 1 & 5 \end{array}\right]+\left[\begin{array}{cc} 14 & 6 \\ -1 & 15 \end{array}\right]} \\ & =\left[\begin{array}{cc} 2+14 & 8+6 \\ 1+(-1) & 5+15 \end{array}\right] \\ & =\left[\begin{array}{cc} 16 & 14 \\ 0 & 20 \end{array}\right] \end{aligned} $$
Soalan 4:
$$ \text { Diberi }\left[\begin{array}{c} 3 a+2 \\ 9-b \end{array}\right]+\left[\begin{array}{c} 4 \\ 2 b \end{array}\right]=\left[\begin{array}{c} 15 \\ -8 \end{array}\right] \text {, hitung nilai } a \text { dan nilai } b \text {. } $$
Penyelesaian:
$$ \begin{aligned} {\left[\begin{array}{c} 3 a+2 \\ 9-b \end{array}\right]+\left[\begin{array}{c} 4 \\ 2 b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \\ {\left[\begin{array}{c} 3 a+2+4 \\ 9-b+2 b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \\ {\left[\begin{array}{c} 3 a+6 \\ 9+b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} 3 a+6 & =15 \\ 3 a & =9 \\ a & =\frac{9}{3} \\ a & =3 \end{aligned} $$
$$ \begin{aligned} 9+b & =-8 \\ b & =-8-9 \\ b & =-17 \end{aligned} $$
$$ \text { Diberi }\left[\begin{array}{c} 3 a+2 \\ 9-b \end{array}\right]+\left[\begin{array}{c} 4 \\ 2 b \end{array}\right]=\left[\begin{array}{c} 15 \\ -8 \end{array}\right] \text {, hitung nilai } a \text { dan nilai } b \text {. } $$
Penyelesaian:
$$ \begin{aligned} {\left[\begin{array}{c} 3 a+2 \\ 9-b \end{array}\right]+\left[\begin{array}{c} 4 \\ 2 b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \\ {\left[\begin{array}{c} 3 a+2+4 \\ 9-b+2 b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \\ {\left[\begin{array}{c} 3 a+6 \\ 9+b \end{array}\right] } & =\left[\begin{array}{l} 15 \\ -8 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} 3 a+6 & =15 \\ 3 a & =9 \\ a & =\frac{9}{3} \\ a & =3 \end{aligned} $$
$$ \begin{aligned} 9+b & =-8 \\ b & =-8-9 \\ b & =-17 \end{aligned} $$