Determinants

Due by Mon, 29 Jan 2018

1. If $det \begin{pmatrix}x & y & z \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{pmatrix} = 1$, compute the determinants of each of the following matrices:
   • (a) $\begin{pmatrix} 2x & 2y & 2z \\ \frac{3}{2} & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}$.
   • (b) $\begin{pmatrix} x & y & z \\ 3x+3 & 3y & 3z+2 \\ x + 1 & y + 1 & z + 1 \end{pmatrix}$.
   • (c) $\begin{pmatrix} x - 1 & y - 1 & z - 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{pmatrix}$.

2. Show that $det \begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{pmatrix} = (b - a)(c - a)(c - b)$.

3. Let $U$ and $V$ be upper-triangular matrices.
   • (a) Show that $U+ V$ is upper-triangular.
   • (b) Prove or disprove: $det(UV) = det(U)det(V)$.
   • (c) Prove or disprove: $det(U+V) = det(U) + det(V)$.

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