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$A_2_times_2=[a_i_j]=ij$

$B_2_times_2=[b_i_j]=i-j$ dan

$C_2_times_2=[c_i_j]=~|i-j|$
pernyataan berikut yang BENAR adalah?
(1) Jika A+B=C+D maka $D_2_times_2=[d_i_j]=ij$

(2) Jika AB=XC maka $X=[x_i_j]=-(ij)$

(3) B tidak mempunyai invers

(4) A matriks singular

Diketahui

Jawaban Terkonfirmasi

Dengan elemen yang dibuat dengan rumus berikut, berlaku:

$a_{11}=1times1=1 \ a_{12}=1times2=2 \ a_{21}=2times1=2 \ a_{22}=2times2=4$

Memberikan:

$A= left[begin{array}{cc}a_{11}&a_{12}\a_{21}&a_{22}end{array}right]=left[begin{array}{cc}1&2\2&4end{array}right]$

Lanjutkan.

$b_{11}=1-1=0 \ b_{12}=1-2=-1 \ b_{21}=2-1=1 \ b_{22}=2-2=0$

Memberikan:

$B= left[begin{array}{cc}b_{11}&b_{12}\b_{21}&b_{22}end{array}right]=left[begin{array}{cc}0&-1\1&0end{array}right]$

Satu lagi:

$c_{11}=|1-1|=0 \ c_{12}=|1-2|=1 \ c_{21}=|2-1|=1 \ c_{22}=|2-2|=0$

Didapat:

$C= left[begin{array}{cc}c_{11}&c_{12}\c_{21}&c_{22}end{array}right]=left[begin{array}{cc}0&1\1&0end{array}right]$

Maka

Pernyataan 1.

$$begin{align}A+B&=C+D \ D&=A+B-C \ D&=left[begin{array}{cc}1&2\2&4end{array}right]+left[begin{array}{cc}0&-1\1&0end{array}right]-left[begin{array}{cc}0&1\1&0end{array}right] \ D&=left[begin{array}{cc}1&0\2&4end{array}right]end{align}$

Cek pada pola matriks D, tampak tidak berlaku karena berbeda dengan matriks A.

Pernyataan 2.

$$begin{align}AB&=XC \ X&=ABC^{-1} \ X&=left[begin{array}{cc}1&2\2&4end{array}right]left[begin{array}{cc}0&-1\1&0end{array}right]timesleft[begin{array}{cc}0&1\1&0end{array}right]^{-1} \ X&=left[begin{array}{cc}2&-1\4&-2end{array}right]left[begin{array}{cc}0&1\1&0end{array}right] \ X&=left[begin{array}{cc}-1&2\-2&4end{array}right]end{align}$

Jika dicermati, seharusnya membentuk:

$D_{2times2}=[D_{ij}]=(-1)^jtimes ij$

Pernyataan 3.
Determinan B memiliki nilai:
det B = 0.0 – 1 (-1)
det B = 1
Matriks B memiliki invers.

Pernyataan 4.
Cek determinan:
det A = 1.4 – 2.2
det A = 0
Matriks A singular.