Paper 1, Section II, 8C8 \mathbf{C}

Vectors and Matrices | Part IA, 2016

(a) Show that the equations

1+s+t=a1s+t=b12t=c\begin{array}{r} 1+s+t=a \\ 1-s+t=b \\ 1-2 t=c \end{array}

determine ss and tt uniquely if and only if a+b+c=3a+b+c=3.

Write the following system of equations

5x+2yz=1+s+t2x+5yz=1s+txy+8z=12t\begin{aligned} &5 x+2 y-z=1+s+t \\ &2 x+5 y-z=1-s+t \\ &-x-y+8 z=1-2 t \end{aligned}

in matrix form Ax=bA \mathbf{x}=\mathbf{b}. Use Gaussian elimination to solve the system for x,yx, y, and zz. State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of AA ?

For which values of x,yx, y, and zz is it possible to solve the above system for ss and tt ?

(b) Define a unitary n×nn \times n matrix. Let AA be a real symmetric n×nn \times n matrix, and let II be the n×nn \times n identity matrix. Show that (A+iI)x2=Ax2+x2|(A+i I) \mathbf{x}|^{2}=|A \mathbf{x}|^{2}+|\mathbf{x}|^{2} for arbitrary xCn\mathbf{x} \in \mathbb{C}^{n}, where x2=j=1nxj2|\mathbf{x}|^{2}=\sum_{j=1}^{n}\left|x_{j}\right|^{2}. Find a similar expression for (AiI)x2|(A-i I) \mathbf{x}|^{2}. Prove that (AiI)(A+iI)1(A-i I)(A+i I)^{-1} is well-defined and is a unitary matrix.

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