Paper 1, Section II, B

Vectors and Matrices | Part IA, 2009

Explain why the number of solutions xR3\mathbf{x} \in \mathbb{R}^{3} of the matrix equation Ax=cA \mathbf{x}=\mathbf{c} is 0,1 or infinity, where AA is a real 3×33 \times 3 matrix and cR3\mathbf{c} \in \mathbb{R}^{3}. State conditions on AA and c\mathbf{c} that distinguish between these possibilities, and state the relationship that holds between any two solutions when there are infinitely many.

Consider the case

A=(aabbaaaba) and c=(1c1)A=\left(\begin{array}{lll} a & a & b \\ b & a & a \\ a & b & a \end{array}\right) \quad \text { and } \mathbf{c}=\left(\begin{array}{l} 1 \\ c \\ 1 \end{array}\right)

Use row and column operations to find and factorize the determinant of AA.

Find the kernel and image of the linear map represented by AA for all values of aa and bb. Find the general solution to Ax=cA \mathbf{x}=\mathbf{c} for all values of a,ba, b and cc for which a solution exists.

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