2.10: Supplementary Exercises for Chapter 2

Supplementary Exercises for Chapter [chap:2]

solutions

2

Solve for the matrix \(X\) if:

\(PXQ = R\); \(XP = S\);

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

\[p(X) = X^{3} - 5X^{2} + 11X - 4I. \nonumber \]

1. If \(p(U) = \left[ \begin{array}{rr} 1 & 3 \\ -1 & 0 \end{array} \right]\) compute \(p(U^{T})\).

2. If \(p(U) = 0\) where \(U\) is \(n \times n\), find \(U^{-1}\) in terms of \(U\).

2. \(U^{-1} = \frac{1}{4}(U^{2} - 5U + 11I)\).

Show that, if a (possibly nonhomogeneous) system of equations is consistent and has more variables than equations, then it must have infinitely many solutions. [Hint: Use Theorem [thm:002811] and Theorem [thm:001473].]

Assume that a system \(A\mathbf{x} = \mathbf{b}\) of linear equations has at least two distinct solutions \(\mathbf{y}\) and \(\mathbf{z}\).

1. Show that \(\mathbf{x}_{k} = \mathbf{y} + k(\mathbf{y} - \mathbf{z})\) is a solution for every \(k\).
2. Show that \(\mathbf{x}_{k} = \mathbf{x}_{m}\) implies \(k = m\). [Hint: See Example [exa:002159].]
3. Deduce that \(A\mathbf{x} = \mathbf{b}\) has infinitely many solutions.

2. If \(\mathbf{x}_{k} = \mathbf{x}_{m}\), then \(\mathbf{y} + k(\mathbf{y} - \mathbf{z}) = \mathbf{y} + m(\mathbf{y} - \mathbf{z})\). So \((k - m)(\mathbf{y} - \mathbf{z}) = \mathbf{0}\). But \(\mathbf{y} - \mathbf{z}\) is not zero (because \(\mathbf{y}\) and \(\mathbf{z}\) are distinct), so \(k - m = 0\) by Example [exa:002159].

1. Let \(A\) be a \(3 \times 3\) matrix with all entries on and below the main diagonal zero. Show that \(A^{3} = 0\).
2. Generalize to the \(n \times n\) case and prove your answer.

[ex:ex2_suppl_6] Let \(I_{pq}\) denote the \(n \times n\) matrix with \((p, q)\)-entry equal to \(1\) and all other entries \(0\). Show that:

1. \(I_{n} = I_{11} + I_{22} + \cdots + I_{nn}\).

2. \(I_{pq}I_{rs} = \left\lbrace \begin{array}{cl} I_{ps} & \mbox{if } q = r \\ 0 & \mbox{if } q \neq r \end{array} \right.\).

3. If \(A = \left[ a_{ij} \right]\) is \(n \times n\), then \(A = \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij}I_{ij}\).
4. If \(A = \left[ a_{ij} \right]\), then \(I_{pq}AI_{rs} = a_{qr}I_{ps}\) for all \(p\), \(q\), \(r\), and \(s\).

4. Using parts (c) and (b) gives \(I_{pq}AI_{rs} = \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij}I_{pq}I_{ij}I_{rs}\). The only nonzero term occurs when \(i = q\) and \(j = r\), so \(I_{pq}AI_{rs} = a_{qr}I_{ps}\).

A matrix of the form \(aI_{n}\), where \(a\) is a number, is called an \(n \times n\) scalar matrix.

1. Show that each \(n \times n\) scalar matrix commutes with every \(n \times n\) matrix.
2. Show that \(A\) is a scalar matrix if it commutes with every \(n \times n\) matrix. [Hint: See part (d.) of Exercise [ex:ex2_suppl_6].]

2. If \(A = \left[a_{ij}\right] = \sum_{ij}a_{ij}I_{ij}\), then \(I_{pq}AI_{rs} = a_{qr}I_{ps}\) by 6(d). But then \(a_{qr}I_{ps} = AI_{pq}I_{rs} = 0\) if \(q \neq r\), so \(a_{qr} = 0\) if \(q \neq r\). If \(q = r\), then \(a_{qq}I_{ps} = AI_{pq}I_{rs} = AI_{ps}\) is independent of \(q\). Thus \(a_{qq} = a_{11}\) for all \(q\).

Let \(M = \left[ \begin{array}{rr} A & B \\ C & D \end{array} \right]\), where \(A\), \(B\), \(C\), and \(D\) are all \(n \times n\) and each commutes with all the others. If \(M^{2} = 0\), show that \((A + D)^{3} = 0\). [Hint: First show that \(A^{2} = -BC = D^{2}\) and that

\[B(A + D) = 0 = C(A + D).] \nonumber \]

If \(A\) is \(2 \times 2\), show that \(A^{-1} = A^{T}\) if and only if \(A = \left[ \begin{array}{rr} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right]\) for some \(\theta\) or \(A = \left[ \begin{array}{rr} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array} \right]\) for some \(\theta\).

[Hint: If \(a^{2} + b^{2} = 1\), then \(a = \cos \theta\), \(b = \sin \theta\) for some \(\theta\). Use

\[\cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi.] \nonumber \]

1. If \(A = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]\), show that \(A^{2} = I\).

2. 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 \(EF = FE\).

If \(A\) is a \(2 \times 2\) real matrix, \(A^{2} = A\) and \(A^{T} = A\), show that either \(A\) is one of \(\left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\),
\(\left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right]\), \(\left[ \begin{array}{rr} 0 & 0 \\ 0 & 1 \end{array} \right]\), \(\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]\), or \(A = \left[ \begin{array}{cc} a & b \\ b & 1 - a \end{array} \right]\) where \(a^{2} + b^{2} = a\), \(-\frac{1}{2} \leq b \leq \frac{1}{2}\) and \(b \neq 0\).

Show that the following are equivalent for matrices \(P\), \(Q\):

1. \(P\), \(Q\), and \(P + Q\) are all invertible and

   \[(P + Q)^{-1} = P^{-1} + Q^{-1} \nonumber \]

2. \(P\) is invertible and \(Q = PG\) where \(G^{2} + G + I = 0\).