Consider the following system of equations :
x + 2y $-$ 3z = a
2x + 6y $-$ 11z = b
x $-$ 2y + 7z = c,
where a, b and c are real constants. Then the system of equations :
Solution
$$D = \left| {\matrix{
1 & 2 & { - 3} \cr
2 & 6 & { - 11} \cr
1 & { - 2} & 7 \cr
} } \right|$$<br><br>= 20 $-$ 2(25) $-$3($-$10)<br><br>= 20 $-$ 50 + 30 = 0<br><br>$${D_1} = \left| {\matrix{
a & 2 & { - 3} \cr
b & 6 & { - 11} \cr
c & { - 2} & 7 \cr
} } \right|$$<br><br>= 20a $-$ 2(7b + 11c) $-$3($-$2b $-$ 6c)<br><br>= 20a $-$ 14b $-$ 22c + 6b +18c<br><br>= 20a $-$ 8b $-$ 4c<br><br>= 4(5a $-$ 2b $-$ c)<br><br>$${D_2} = \left| {\matrix{
1 & a & { - 3} \cr
2 & b & { - 11} \cr
1 & c & 7 \cr
} } \right|$$<br><br>= 7b + 11c $-$ a(25) $-$3(2c $-$ b)<br><br>= 7b + 11c $-$ 25a $-$ 6c + 3b<br><br>= $-$25a + 10b + 5c<br><br>= $-$5(5a $-$ 2b $-$ c)<br><br>$${D_3} = \left| {\matrix{
1 & 2 & a \cr
2 & 6 & b \cr
1 & { - 2} & c \cr
} } \right|$$<br><br>= 6c + 2b $-$ 2(2c $-$ b) $-$ 10a<br><br>= $-$10a + 4b + 2c<br><br>= $-$2(5a $-$ 2b $-$ c)<br><br>for infinite solution <br><br>$D = {D_1} = {D_2} = {D_3} = 0$<br><br>$\Rightarrow$ 5a = 2b + c
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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