A=⎝⎛102−1231−10⎠⎞,X=⎝⎛xyz⎠⎞,B=⎝⎛111⎠⎞
AX=B
A−1AX=A−1B=>X=A−1B augment the matrix with the identity matrix:
⎝⎛102−1231−10100010001⎠⎞ R3=R3−2R1
⎝⎛100−1251−1−210−2010001⎠⎞ R2=R2/2
⎝⎛100−1151−1/2−210−201/20001⎠⎞ R1=R1+R2
⎝⎛1000151/2−1/2−210−21/21/20001⎠⎞ R3=R3−5R2
⎝⎛1000101/2−1/21/210−21/21/2−5/2001⎠⎞ R3=2R3
⎝⎛1000101/2−1/2110−41/21/2−5002⎠⎞ R1=R1−R3/2
⎝⎛1000100−1/2130−431/2−5−102⎠⎞ R2=R2+R3/2
⎝⎛1000100013−2−43−2−5−112⎠⎞We are done. On the left is the identity matrix. On the right is the inverse matrix.
A−1=⎝⎛3−2−43−2−5−112⎠⎞
X=A−1B
=⎝⎛3−2−43−2−5−112⎠⎞⎝⎛111⎠⎞
=⎝⎛3+3−1−2−2+1−4−5+2⎠⎞=⎝⎛5−3−7⎠⎞
(5,−3,−7)
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