Answer:
b) inconsistent
d) inconsistent
f) inconsistent
j) X_1 = 2 - 6a , X_2 = 4 + a , X_3 = 3 - a, X_4 = a (free variable)
Step-by-step explanation:
b)
Given: Augmented matrix
![\left[\begin{array}{ccc}2&-3&5\\-4&6&8\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-3%265%5C%5C-4%266%268%5Cend%7Barray%7D%5Cright%5D)
Row operation : R_2 + 2*R_1
![\left[\begin{array}{ccc}2&-3&5\\0&0&18\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-3%265%5C%5C0%260%2618%5Cend%7Barray%7D%5Cright%5D)
Looking at R_2 : 0 not equal to 18 ; Hence, inconsistent
c)
Given: Augmented matrix
![\left[\begin{array}{cccc}3&2&-3&4\\1&-2&2&1\\11&2&1&14\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D3%262%26-3%264%5C%5C1%26-2%262%261%5C%5C11%262%261%2614%5Cend%7Barray%7D%5Cright%5D)
Row operation : R_2 <---> R_1
![\left[\begin{array}{cccc}1&-2&2&1\\3&2&-3&4\\11&2&1&14\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-2%262%261%5C%5C3%262%26-3%264%5C%5C11%262%261%2614%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_2 - 3*R_1 & R_3 - 11*R_1
![\left[\begin{array}{cccc}1&-2&2&1\\0&8&-9&1\\0&24&-21&3\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-2%262%261%5C%5C0%268%26-9%261%5C%5C0%2624%26-21%263%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_3 - 3*R_2
![\left[\begin{array}{cccc}1&-2&2&1\\0&8&-9&1\\0&0&6&0\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-2%262%261%5C%5C0%268%26-9%261%5C%5C0%260%266%260%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Looking at R_3 : 0 not equal to 6 ; Hence, inconsistent
f)
Given: Augmented matrix
![\left[\begin{array}{cccc}1&-1&2&4\\2&3&-1&1\\7&3&4&7\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-1%262%264%5C%5C2%263%26-1%261%5C%5C7%263%264%267%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_2 - 2*R_1 & R_3 - 7*R_1
![\left[\begin{array}{cccc}1&-1&2&4\\0&5&-5&-7\\0&10&-10&-21\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-1%262%264%5C%5C0%265%26-5%26-7%5C%5C0%2610%26-10%26-21%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_2 --> ( 1 / 5 )* R_2 & R_3 --> ( 1 / 10 )* R_3
![\left[\begin{array}{cccc}1&-1&2&4\\0&1&-1&-1.4\\0&1&-1&-2.1\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-1%262%264%5C%5C0%261%26-1%26-1.4%5C%5C0%261%26-1%26-2.1%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_3 - R_1
![\left[\begin{array}{cccc}1&-1&2&4\\0&1&-1&-1.4\\0&0&0&-0.7\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-1%262%264%5C%5C0%261%26-1%26-1.4%5C%5C0%260%260%26-0.7%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Looking at R_3 : 0 not equal to -0.7 ; Hence, inconsistent
j)
Given: Augmented matrix
![\left[\begin{array}{ccccc}1&2&-3&1&1\\-1&-1&4&-1&6\\-2&-4&7&-1&1\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%262%26-3%261%261%5C%5C-1%26-1%264%26-1%266%5C%5C-2%26-4%267%26-1%261%5Cend%7Barray%7D%5Cright%5D%5C%5C)
Row operations : R_2 + R_1 & R_3 + 2*R_1
![\left[\begin{array}{ccccc}1&2&-3&1&1\\0&1&1&0&7\\0&0&1&1&3\end{array}\right]\\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%262%26-3%261%261%5C%5C0%261%261%260%267%5C%5C0%260%261%261%263%5Cend%7Barray%7D%5Cright%5D%5C%5C)
corresponding equations are:
X_1 + 2*X_2 - 3*X_3 + X_4 = 1
X_2 + X_3 = 7
X_3 + X_4 = 3
Free parameter : X_4 = a , then:
X_3 = 3 - a
X_2 = 4 + a
X_1 = 1 - a + 3*(3 - a) - 2*(4 + a) = 2 - 6 a