Find the values of a and b that make the equation system have infinite solutions

Question

I'm given the following system of equations and I need to find the values for a and b so that the system has infinite solutions. \begin{cases} &3y &- &2z &= 5 \\ &-2x &+ &az &= -7 \\ &3x &- &2y &= b \end{cases}

I know that in order for the system to have infinite solutions the rank of the matrix of the coefficents should be the same as the rank of the augmented matrix and lower than the number of equations, but I'm not sure how to find the values of a and b that satisfy such a condition.

Answer 1

The augmented matrix $[A\mid b~]$ is$$\begin{vmatrix}0&3&-2&|&5\\-2&0&a&|&-7\\3&-2&0&|&b\end{vmatrix}$$After reducing $A$ to row-echelon form:$$\begin{vmatrix}3&-2&0&|&b\\0&3&-2&|&5\\0&0&a-8/9&|&2b/3-43/9\end{vmatrix}$$Now, $\text{rank}(A)=\text{rank}([A\mid b~])<3\implies a=8/9,b=43/6$.

Answer 2

Multiplying 2. with 3 and 3. equation with 2 and add them we get $$3az-4y = -21+2b$$ Multiplying this one with 3 and 1. one with 4 we get: $$-8z+9az=20-63+6b$$

and thus $$ z(9a-8)= 6b-43$$ So if

• $ a = {8\over 9}$ and $b={43\over 6}$ we have infinite solutions.
• $ a = {8\over 9}$ and $b\ne {43\over 6}$ we have no solution.
• $ a \ne {8\over 9}$ and $b\in\mathbb{R}$ we have one solution.

Answer 3

For infinite solution there are two conditions

1. $N(A)\neq\{0\}$

2. $b\in Col(A)$