Question

Determine all values of the constant a for which the following system has (a) no
solution, (b) an infinite number of solutions, and (c) a unique solution.
ax1 + x2 + x3 = 1,
x1 + ax2 + x3 = 1,
x1 + x2 + ax3 = 1.

Answer 1

Answer on Question #75616 – Math – Linear Algebra

Question

Determine all values of the constant $a$ for which the following system has (a) no solution, (b) an infinite number of solutions, and (c) a unique solution.

$$a x _ {1} + x _ {2} + x _ {3} = 1,$$ $$x _ {1} + a x _ {2} + x _ {3} = 1,$$ $$x _ {1} + x _ {2} + a x _ {3} = 1.$$

Solution

The coefficient matrix

$$A = \left( \begin{array}{ccc} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \end{array} \right)$$

Find

$$\begin{array}{l} det A = \left| \begin{array}{ccc} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \end{array} \right| = a \left| \begin{array}{cc} a & 1 \\ 1 & a \end{array} \right| - \left| \begin{array}{cc} 1 & 1 \\ 1 & a \end{array} \right| + \left| \begin{array}{cc} 1 & a \\ 1 & 1 \end{array} \right| = \\ = a (a ^ {2} - a) - (a - 1) + (1 - a) = (a - 1) (a (a + 1) - 2) = \\ = (a - 1) ^ {2} (a + 2) \end{array}$$

The linear system has a unique solution iff $detA\neq 0$

$$(a - 1) ^ {2} (a + 2) \neq 0$$ $$a \neq - 2, a \neq 1$$

If $a = 1$

$$x _ {1} + x _ {2} + x _ {3} = 1,$$ $$x _ {1} + x _ {2} + x _ {3} = 1,$$ $$x _ {1} + x _ {2} + x _ {3} = 1.$$

There is one equation for three variables.

The system has an infinite number of solutions.

If $a = -2$

$$- 2 x _ {1} + x _ {2} + x _ {3} = 1,$$ $$x _ {1} - 2 x _ {2} + x _ {3} = 1,$$ $$x _ {1} + x _ {2} - 2 x _ {3} = 1.$$

Add three equations

$$\left(- 2 x _ {1} + x _ {2} + x _ {3}\right) + \left(x _ {1} - 2 x _ {2} + x _ {3}\right) + \left(x _ {1} + x _ {2} - 2 x _ {3}\right) = 1 + 1 + 1$$

We have

$$0 = 3, False$$

The system is inconsistent. Therefore, the system has no solutions.

Answer:

(a) If $a = -2$, the system has no solution.

(b) If $a = 1$, the system has an infinite number of solutions.

(c) If $a \neq -2, a \neq 1$ the system has a unique solution.

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