Question

Use the augmented matrix to determine if the linear system is consistent. Is the linear system represented by the augmented matrix​ consistent? A. ​Yes, because the rightmost column of the augmented matrix is a pivot column. B. ​Yes, because the rightmost column of the augmented matrix is not a pivot column. C. ​No, because the rightmost column of the augmented matrix is a pivot column. D. ​No, because the rightmost column of the augmented matrix is not a pivot column.

Answer 1

Answer:

The correct option is (A).

Step-by-step explanation:

If the reduced row echelon form of the coefficient matrix of a linear system of equations in four different variables has a pivot, i.e. 1, in each column, then the reduced row echelon form of the coefficient matrix is say A is an identity matrix, here I₄, since there are 4 variables.

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]$

Then the corresponding augmented matrix [ A|B] , where the matrix is the representation of the linear system is AX = B, must be:

$\left[\begin{array}{ccccc}1&0&0&0&a\\0&1&0&0&b\\0&0&1&0&c\\0&0&0&1&d\end{array}\right]$

Now the given linear system is consistent as the right most column of the augmented matrix is a linear combination of the columns of A as the reduced row echelon form of A has a pivot in each column.

Thus, the correct option is (A).