MATRIKS 1. SPMU Diberi bahawa matriks $$ P=\left|\begin{array}{cc}8 & h \ 4 & -3\end{array} ight| $$. It is given that matrix $$ P=\left[\begin{array}{cc}8 & h \ 4 & -3\end{array} ight] $$. (a) Jika matriks $$ P $$ tidak mempunyai songsangan, tentukan nilai $$ h $$. [2 martah] If matrix $$ P $$ has no inverse, determine the value of $$ h $$. [2 marks] (b) Seterusnya, cari matriks $$ M $$ jika $$ P^{2}+M=3 P $$. [3 martah] Hence, find matrix $$ M $$ if $$ P^{2}+M=3 P $$. [ 3 merist] Jawapan / Answer: (a) (b)

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MATRIKS 1. SPMU Diberi bahawa matriks $$ P=\left|\begin{array}{cc}8 & h \ 4 & -3\end{array}ight| $$. It is given that matrix $$ P=\left[\begin{array}{cc}8 & h \ 4 & -3\end{array}ight] $$. (a) Jika matriks $$ P $$ tidak mempunyai songsangan, tentukan nilai $$ h $$. [2 martah] If matrix $$ P $$ has no inverse, determine the value of $$ h $$. [2 marks] (b) Seterusnya, cari matriks $$ M $$ jika $$ P^{2}+M=3 P $$. [3 martah] Hence, find matrix $$ M $$ if $$ P^{2}+M=3 P $$. [ 3 merist] Jawapan / Answer: (a) (b)

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**Jawapan / Answer:**

**(a) Menentukan nilai $$ h $$ apabila matriks $$ P $$ tidak mempunyai songsangan**

Matriks $$ P $$ tidak mempunyai songsangan jika determinannya adalah sifar. Determinan matriks $$ P $$ adalah:

$$
\det(P) = (8)(-3) - (4)(h) = -24 - 4h
$$

Untuk menjadikan determinan sama dengan sifar:

$$
-24 - 4h = 0 \
-4h = 24 \
h = -6
$$

**Nilai $$ h $$ ialah $$ -6 $$.**

---

**(b) Mencari matriks $$ M $$ jika $$ P^{2} + M = 3P $$**

Diberi $$ h = -6 $$, matriks $$ P $$ menjadi:

$$
P = \begin{bmatrix} 8 & -6 \ 4 & -3 \end{bmatrix}
$$

Pertama, kita perlu mengira $$ P^{2} $$:

$$
P^{2} = P \times P = \begin{bmatrix} 8 & -6 \ 4 & -3 \end{bmatrix} \times \begin{bmatrix} 8 & -6 \ 4 & -3 \end{bmatrix}
$$

$$
P^{2} = \begin{bmatrix}
(8 \times 8) + (-6 \times 4) & (8 \times -6) + (-6 \times -3) \
(4 \times 8) + (-3 \times 4) & (4 \times -6) + (-3 \times -3)
\end{bmatrix} = \begin{bmatrix}
64 - 24 & -48 + 18 \
32 - 12 & -24 + 9
\end{bmatrix} = \begin{bmatrix}
40 & -30 \
20 & -15
\end{bmatrix}
$$

Kemudian, kita gunakan persamaan $$ P^{2} + M = 3P $$ untuk mencari $$ M $$:

$$
3P = 3 \times \begin{bmatrix} 8 & -6 \ 4 & -3 \end{bmatrix} = \begin{bmatrix} 24 & -18 \ 12 & -9 \end{bmatrix}
$$

$$
M = 3P - P^{2} = \begin{bmatrix} 24 & -18 \ 12 & -9 \end{bmatrix} - \begin{bmatrix} 40 & -30 \ 20 & -15 \end{bmatrix} = \begin{bmatrix} 24-40 & -18-(-30) \ 12-20 & -9-(-15) \end{bmatrix} = \begin{bmatrix} -16 & 12 \ -8 & 6 \end{bmatrix}
$$

**Matriks $$ M $$ adalah:**

$$
M = \begin{bmatrix} -16 & 12 \ -8 & 6 \end{bmatrix}
$$
**(a) Nilai $$ h $$ adalah $$ -6 $$.** **(b) Matriks $$ M $$ adalah:** $$ M = \begin{bmatrix} -16 & 12 \ -8 & 6 \end{bmatrix} $$

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