Challenge Problems for Chapter 4

Solve for the matrix \(X\) if:
(a)
\(PXQ = R\);
(b)
\(XP = S\);

where \( P = \left [ \begin{array}{rr} 1 & 0 \\ 2 & -1 \\ 0 & 3 \end{array} \right ]\), \( Q = \left [ \begin{array}{rrr} 1 & 1 & -1 \\ 2 & 0 & 3 \end{array} \right ]\),
\( R = \left [ \begin{array}{rrr} -1 & 1 & -4 \\ -4 & 0 & -6 \\ 6 & 6 & -6 \end{array} \right ]\), \( S = \left [ \begin{array}{rr} 1 & 6\\ 3 & 1 \end{array} \right ]\)

Consider
\begin{equation*} p(X) = X^{3} - 5X^{2} + 11X - 4I. \end{equation*}
(a)
If \(p(U) = \left [ \begin{array}{rr} 1 & 3 \\ -1 & 0 \end{array} \right ]\) compute \(p(U^{T})\).
(b)
If \(p(U) = 0\) where \(U\) is \(n \times n\), find \(U^{-1}\) in terms of \(U\).

Click on the arrow to see answer.

\(U^{-1} = \frac {1}{4}(U^{2} - 5U + 11I)\).
Assume that a system \(A\vec {x} = \vec {b}\) of linear equations has at least two distinct solutions \(\vec {y}\) and \(\vec {z}\).
(a)
Show that \(\vec {x}_{k} = \vec {y} + k(\vec {y} - \vec {z})\) is a solution for every \(k\).
(b)
Show that \(\vec {x}_{k} = \vec {x}_{m}\) implies \(k = m\).
(c)
Deduce that \(A\vec {x} = \vec {b}\) has infinitely many solutions.

Click on the arrow to see answer.

(a)
If \(\vec {x}_{k} = \vec {x}_{m}\), then \(\vec {y} + k(\vec {y} - \vec {z}) = \vec {y} + m(\vec {y} - \vec {z})\). So \((k - m)(\vec {y} - \vec {z}) = \vec {0}\). But \(\vec {y} - \vec {z}\) is not zero (because \(\vec {y}\) and \(\vec {z}\) are distinct), so \(k - m = 0\).
Let \(A = \left [ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right ]\)
(a)
show that \(A^{2} = I\).
(b)
What is wrong with the following argument? If \(A^{2} = I\), then \(A^{2} - I = 0\), so \((A - I)(A + I) = 0\), whence \(A = I\) or \(A = -I\).
Let \(E\) and \(F\) be elementary matrices obtained from the identity matrix by adding multiples of row \(k\) to rows \(p\) and \(q\). If \(k \neq p\) and \(k \neq q\), show that \(EF = FE\).

Bibliography

These exercises come from the end of Chapter 2 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 143–144.