Mathematical Immaturity

3.6 $\quad$ Exercises

$\quad$ In this set of exercises you may assume existence of a determinant function. Determinants of order three may be computed by Equation (3.2).

\begin{align*} (3.2) \qquad \det\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} &= a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix} \end{align*}

• Compute each of the following determinants. \begin{align*} \text{(a)} \quad \begin{vmatrix} 2 & 1 & 1 \\ 1 & 4 & -4 \\ 1 & 0 & 2 \end{vmatrix}, \quad \text{(b)} \quad \begin{vmatrix} 3 & 0 & 8 \\ 5 & 0 & 7 \\ -1 & 4 & 2 \end{vmatrix}, \quad \text{(c)} \quad \begin{vmatrix} a & 1 & 0 \\ 2 & a & 2 \\ 0 & 1 & a \end{vmatrix}. \end{align*}
• If det $\begin{bmatrix} x & y & z \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{bmatrix} = 1,$ compute the determinant of each of the following matrices: \begin{align*} \text{(a)} \quad \begin{bmatrix} 2x & 2y & 2z \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{bmatrix}, \quad \text{(b)} \quad \begin{bmatrix} x & y & z \\ 3x + 3 & 3y & 3z + 2 \\ x + 1 & y + 1 & z + 1 \end{bmatrix}, \quad \text{(c)} \quad \begin{bmatrix} x - 1 & y - 1 & z - 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix}. \end{align*}
• (a) Prove that $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (b - a)(c - a)(c - b).$
  (b) Find corresponding formulas for the determinants $$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} \quad \text{and} \quad \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}.$$
• Compute the determinant of each of the following matrices by transforming each of them to an upper triangular matrix. \begin{align} &\text{(a)} \qquad \begin{bmatrix} 1 & -1 & 1 & 1 \\ 1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & 1 & 1 & -1 \end{bmatrix} \\ \\ &\text{(b)} \qquad \begin{bmatrix} 1 & 1 & 1 & 1 \\ a & b & c & d \\ a^2 & b^2 & c^2 & d^2 \\ a^3 & b^3 & c^3 & d^3 \end{bmatrix} \\ \\ &\text{(c)} \qquad \begin{bmatrix} 1 & 1 & 1 & 1 \\ a & b & c & d \\ a^2 & b^2 & c^2 & d^2 \\ a^4 & b^4 & c^4 & d^4 \end{bmatrix} \\ \\ &\text{(d)} \qquad \begin{bmatrix} a & 1 & 0 & 0 & 0 \\ 4 & a & 2 & 0 & 0 \\ 0 & 3 & a & 3 & 0 \\ 0 & 0 & 2 & a & 4 \\ 0 & 0 & 0 & 1 & a \end{bmatrix} \\ \\ &\text{(e)} \qquad \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 & 1 & -1 \end{bmatrix} \end{align}
• A lower triangular matrix $A = (a_{ij})$ is a square matrix with all entries above the main diagonal equal to 0; that is, $a_{ij} = 0$ whenever $i \lt j.$ Prove that the determinant of such a matrix is equal to the product of its diagonal entries: det $A = a_{11}a_{22} \cdots a_{nn}.$
• Let $f_1, f_2, g_1, g_2$ be four functions differentiable on an interval $(a, b).$ Define $$F(x) = \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix}$$ for each $x$ in $(a, b).$ Prove that $$F'(x) = \begin{vmatrix} f_1'(x) & f_2'(x) \\ g_1(x) & g_2(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & f_2(x) \\ g_1'(x) & g_2'(x) \end{vmatrix}.$$
• State and prove a generalization of Exercise 6 for the determinant $$F(x) = \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ g_1(x) & g_2(x) & g_3(x) \\ h_1(x) & h_2(x) & h_3(x) \end{vmatrix}.$$
• (a) $\quad$ If $F(x) = \begin{vmatrix} f_1(x) & f_2(x) \\ f_1'(x) & f_2'(x) \end{vmatrix},$ prove that $F'(x) = \begin{vmatrix} f_1(x) & f_2(x) \\ f_1''(x) & f_2''(x) \end{vmatrix}.$
  
  (b) $\quad$ State and prove a corresponding result for $3 \times 3$ determinants, assuming the validity of Equation (3.2).
• Let $U$ and $V$ be two $n \times n$ upper triangular matrices.
  (a) $\quad$ Prove that each of $U + V$ and $UV$ is an upper triangular matrix.
  (b) $\quad$ Prove that det $(UV) = (\text{det } U)(\text{det } V).$
  (c) $\quad$ If det $U \neq 0$ prove that there is an upper triangular matrix $U^{-1}$ such that $UU^{-1} = I,$ and deduce that det $(U^{-1}) = 1/\text{det } U.$
• Calculate det $A,$ det $(A^{-1}),$ and $A^{-1}$ for the following upper triangular matrix: $$A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 2 & 3 \\ 0 & 0 & 0 & 2 \end{bmatrix}.$$