EE C12

Find the characteristic polynomial of the matrix:

\[A = \left[\begin{array}\\1 & 2 & 1 & 0\\ 1 & 0 & 1 & 0\\ 2 & 1 & 1 & 0\\ 3 & 1 & 0 & 1\\ \end{array}\right]\]

Then the characteristic polynomial of \(A\) is the polynomial \(p_{A} (x)\) defined by:

\[p_{A} (x) = det (A -xI{_n})\]

Solve:

\[\begin{align} p_{A} (x) &= det (A -xI{_n})\\ p_{A} (x) &= \left|\begin{array}\\(1-x) & 2 & 1 & 0\\1 & (0-x) & 1 & 0\\2 & 1 & (1-x) & 0\\3 & 1 & 0 & (1-x)\\\end{array} \right|\\ p_{A} (x) &= (1-x)\left|\begin{array}\\-x & 1 & 0\\1 & (1-x) & 0\\1 & 0 & (1-x)\\\end{array}\right| - (2) \left|\begin{array}\\1 & 1 & 0\\2 & (1-x) & 0\\3 & 0 & (1-x)\\\end{array}\right| + (1) \left|\begin{array}\\1 & -x & 0\\2 & 1 & 0\\3 & 1 & (1-x)\\\end{array}\right| -(0) \left|\begin{array}\\1 & -x & 1\\2 & 1 & 1-x\\3 & 1 & 0\\\end{array}\right|\\ \end{align}\]\

\[\begin{align} (1-x)\left|\begin{array}\\-x & 1 & 0\\1 & 1-x & 0\\1 & 0 & 1-x\\\end{array}\right| &=(1-x) [ (-x)(1-x)^{2} - 1+x) ] \\ &=(1-x)[(-x)(1-2x+x^2)-1+x]\\ &=(1-x)[-x+2x^2-x^3-1+x]\\ &=(1-x)(-x^3+2x^2-1)\\ &=-x^3+2x^2-1+x^4-2x^3+x\\ &=-x^3+2x^2-1+x^4-2x^3+x\\ &=(x^4-3x^3+2x^2+x-1)\\ -(2)\left|\begin{array}\\1 & 1 & 0\\2 & (1-x) & 0\\3 & 0 & (1-x)\\\end{array}\right| &= (-2) [(1-x)^2-2(1-x)+0] \\ &= (-2)[1-2x+x^2-2+2x]\\ &= (-2)(x^2-1)\\ &= (-2x^2+2)\\ +(1) \left|\begin{array}\\1 & -x & 0\\2 & 1 & 0\\3 & 1 & (1-x)\\\end{array}\right| &= (1)(1-x) - (-x)[2(1-x)]\\ &= (1-x) + x(2-2x)\\ &= (1-x) + 2x-2x^2\\ &= (-2x^2+x+1)\\ -(0) \left|\begin{array}\\1 & -x & 1\\2 & 1 & 1-x\\3 & 1 & 0\\\end{array}\right| &= 0 \end{align}\]

\[\begin{align} p_{A}(x) &= (x^4-3x^3+2x^2+x-1) + (-2x^2+2) + (-2x^2+x+1) + (0)\\ &= x^4 - 3x^3 - 2x^2 + 2x + 2 \end{align}\]