Algebra and Geometry | Part IA, 2004

The vector x=(xyz)\mathbf{x}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right) satisfies the equation

Ax=b\mathbf{A} \mathbf{x}=\mathbf{b}

where A\mathbf{A} is a (3×3)(3 \times 3) matrix and b\mathbf{b} is a (3×1)(3 \times 1) column vector. State the conditions under which this equation has (a) a unique solution, (b) an infinity of solutions, (c) no solution for x\mathbf{x}.

Find all possible solutions for the unknowns x,yx, y and zz which satisfy the following equations:

x+y+z=1x+y+λz=2x+2y+λz=4\begin{array}{r} x+y+z=1 \\ x+y+\lambda z=2 \\ x+2 y+\lambda z=4 \end{array}

in the cases (a) λ=0\lambda=0, and (b) λ=1\lambda=1.

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