3.E: Supplementary Exercises for Chapter 3

Exercise \(\PageIndex{1}\)

Show that

\[\det \left[ \begin{array}{ccc} a+px & b+qx & c+rx \\ p+ux & q+vx & r+wx \\ u+ax & v+bx & w+cx \\ \end{array} \right] = (1+x^3) \det \left[ \begin{array}{ccc} a & b & c \\ p & q & r \\ u & v & w \end{array}\right] \nonumber \]

Exercise \(\PageIndex{2}\)

1. Show that \((A_{ij})^{T} = (A^{T})_{ji}\) for all \(i\), \(j\), and all square matrices \(A\).
2. Use (a) to prove that \(\det A^{T} = \det A\). [Hint: Induction on \(n\) where \(A\) is \(n \times n\).]

Answer

2. If \(A\) is \(1 \times 1\), then \(A^{T} = A\). In general, \(\det \left[ A_{ij} \right] = \det \left[ (A_{ij})^{T}\right] = \det \left[(A^{T})_{ji} \right]\) by (a) and induction. Write \(A^{T} = \left[ a^\prime_{ij}\right]\) where \(a^\prime_{ij} = a_{ji}\), and expand \(\det A^{T}\) along column 1. \[\begin{aligned} \det A^T &= \sum_{j=1}^{n} a^{\prime}_{j1}(-1)^{j+1} \det [(A^T)_{j1}] \\ &= \sum_{j=1}^{n} a_{1j}(-1)^{1+j} \det [A_{1j}] = \det A\end{aligned} \nonumber \] where the last equality is the expansion of \(\det A\) along row 1.

Exercise \(\PageIndex{3}\)

Show that \(\det \left[ \begin{array}{cc} 0 & I_n \\ I_m & 0 \end{array}\right] = (-1)^{nm}\) for all \(n \geq 1\) and \(m \geq 1\).

Exercise \(\PageIndex{4}\)

Show that \[\det \left[ \begin{array}{rrr} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{array}\right] = (b-a)(c-a)(c-b)(a+b+c) \nonumber \]

Exercise \(\PageIndex{5}\)

Let \(A = \left[ \begin{array}{c} R_1 \\ R_2 \end{array}\right]\) be a 2 \(\times\) 2 matrix with rows \(R_{1}\) and \(R_{2}\). If \(\det A = 5\), find \(\det B\) where \[B = \left[ \begin{array}{c} 3R_1 + 2R_3 \\ 2R_1 + 5R_2 \end{array}\right] \nonumber \]

Exercise \(\PageIndex{6}\)

Let \(A = \left[ \begin{array}{rr} 3 & -4 \\ 2 & -3 \end{array}\right]\) and let \(\mathbf{v}_{k} = A^{k}\mathbf{v}_{0}\) for each \(k \geq 0\).

1. Show that \(A\) has no dominant eigenvalue.
2. Find \(\mathbf{v}_{k}\) if \(\mathbf{v}_{0}\) equals:

1. \(\left[ \begin{array}{r} 1 \\ 1 \end{array}\right]\)
2. \(\left[ \begin{array}{r} 2 \\ 1 \end{array}\right]\)
3. \(\left[ \begin{array}{r} x \\ y \end{array}\right] \neq \left[ \begin{array}{r} 1 \\ 1 \end{array}\right]\) or \(\left[ \begin{array}{r} 2 \\ 1 \end{array}\right]\)