Question

matematika ...........​

Answer 1

Verified answer

• a. x = 2.
• b. x = –7 atau x = 3.
• c. x = –2 atau x = 1.
• d. x = –2 atau x = –1.
• e. x = 2/3 atau x = –2.
• f. x = ½(3 + i√71) atau x = ½(3 – i√71).

 

Penjelasan dengan langkah-langkah:

Matriks Singular

Matriks singular adalah matriks yang determinannya bernilai 0, sehingga tidak dapat memiliki invers.

Soal a.

Diberikan matriks:

[tex]\begin{aligned}A&=\begin{pmatrix}x & -4\\ 1 & x-4\end{pmatrix}\end{aligned}[/tex]

Matriks A adalah matriks singular, maka:

[tex]\begin{aligned}\det(A)&=0\\x(x-4)-1\cdot(-4)&=0\\x^2-4x+4&=0\\(x-2)^2&=0\\\therefore\ x&=\boxed{\bf2}\end{aligned}[/tex]
__________________

Soal b.

Diberikan matriks:

[tex]\begin{aligned}B&=\begin{pmatrix}7 & x\\ x+4 & 3\end{pmatrix}\end{aligned}[/tex]

Matriks B adalah matriks singular, maka:

[tex]\begin{aligned}\det(B)&=0\\7\cdot3-x(x+4)&=0\\21-x^2-4x&=0\\x^2+4x-21&=0\\(x+7)(x-3)&=0\\\therefore\ x=\boxed{\bf{-7}}{\sf\ atau\ }x&=\boxed{\bf{3}}\end{aligned}[/tex]
__________________

Soal c.

Diberikan matriks:

[tex]\begin{aligned}C&=\begin{pmatrix}2x & 1\\ 4 & x+1\end{pmatrix}\end{aligned}[/tex]

Matriks C adalah matriks singular, maka:

[tex]\begin{aligned}\det(C)&=0\\2x(x+1)-4\cdot1&=0\\2x^2+2x-4&=0\\x^2+x-2&=0\\(x+2)(x-1)&=0\\\therefore\ x=\boxed{\bf{-2}}{\sf\ atau\ }x&=\boxed{\bf{1}}\end{aligned}[/tex]
__________________

Soal d.

Diberikan matriks:

[tex]\begin{aligned}D&=\begin{pmatrix}x+2 & 2x\\ x+2 & x-1\end{pmatrix}\end{aligned}[/tex]

Matriks D adalah matriks singular, maka:

[tex]\begin{aligned}\det(D)&=0\\(x+2)(x-1)-(x+2)(2x)&=0\\(x+2)(x-1-2x)&=0\\(x+2)(-x-1)&=0\\(x+2)(x+1)&=0\\\therefore\ x=\boxed{\bf{-2}}{\sf\ atau\ }x&=\boxed{\bf{-1}}\end{aligned}[/tex]
__________________

Soal e.

Diberikan matriks:

[tex]\begin{aligned}E&=\begin{pmatrix}3x-2 & 3x-2\\ x+1 & -1\end{pmatrix}\end{aligned}[/tex]

Matriks E adalah matriks singular, maka:

[tex]\begin{aligned}\det(E)&=0\\(3x-2)(-1)-(3x-2)(x+1)&=0\\(3x-2)(-1-x-1)&=0\\(3x-2)(-x-2)&=0\\(3x-2)(x+2)&=0\\\therefore\ x=\boxed{\,\bf{\frac{2}{3}}\,}{\sf\ atau\ }x&=\boxed{\bf{-2}}\end{aligned}[/tex]
__________________

Soal f.

Diberikan matriks:

[tex]\begin{aligned}F&=\begin{pmatrix}4 & x+1\\ x & x-5\end{pmatrix}\end{aligned}[/tex]

Matriks F adalah matriks singular, maka:

[tex]\begin{aligned}\det(F)&=0\\4(x-5)-x(x+1)&=0\\4x-20-x^2-x&=0\\3x-20-x^2&=0\\x^2-3x+20&=0\\\end{aligned}[/tex]

Karena tidak bisa difaktorkan, maka kita selesaikan dengan rumus ABC.

[tex]\begin{aligned}x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\&=\frac{-(-3)\pm\sqrt{(-3)^2-4\cdot1\cdot20}}{2\cdot1}\\&=\frac{3\pm\sqrt{9-80}}{2}\\&=\frac{3\pm\sqrt{-71}}{2}\\&=\frac{3\pm\sqrt{-1}\sqrt{71}}{2}\\&=\frac{3\pm i\sqrt{71}}{2}\\\end{aligned}[/tex]

Diperoleh:

[tex]\begin{aligned}x&=\boxed{\bf\frac{3+i\sqrt{71}}{2}}{\sf\ atau\ }x=\boxed{\bf\frac{3-i\sqrt{71}}{2}}\\\end{aligned}[/tex]