Chapter 9

Show that $$ \sum_{k=1}^{n}(-1)^{n-k}\left(\begin{array}{l} n \\ k \end{array}\right) k^{n}=n ! . $$

Let \(f(z)=a_{0}+a_{1} z+\cdots+a_{n} z^{n}\). Find \(z_{0}\) such that the polynomial \(f\left(z+z_{0}\right)\) has no term in \(z^{n-1}\).

Let \(A\) be a \(2 \times 2\) matrix with integer entries. Show that \(A^{-1}\) exists and has integer entries if and only if \(\operatorname{det}(A)=\pm 1\).

Show that $$ \left(\begin{array}{ccc} a & * & * \\ 0 & b & * \\ 0 & 0 & c \end{array}\right)\left(\begin{array}{ccc} a^{\prime} & * & * \\ 0 & b^{\prime} & * \\ 0 & 0 & c^{\prime} \end{array}\right)=\left(\begin{array}{ccc} a a^{\prime} & * & * \\ 0 & b b^{\prime} & * \\ 0 & 0 & c c^{\prime} \end{array}\right), $$ where \(*\) refers to some unspecified entry in the matrix.

Show that if \(A\) is a square matrix, then \(A A^{t}\) is symmetric. Choose any \(2 \times 2\) matrix and verify this directly.

A matrix \(\left(a_{i j}\right)\) is a diagonal matrix if \(a_{i j}=0\) whenever \(i \neq j .\) Show that the space \(D\) of real \(n \times n\) diagonal matrices is a vector space of dimension \(n .\)

Let \(a\) be any non-zero vector in \(\mathbb{R}^{3}\) and let \(\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be the linear map defined by \(\alpha(x)=a \times x\) (the vector product). Without doing any calculations, explain why \(\operatorname{det}(\alpha)=0\).

Show that the set of \(n \times n\) matrices \(A\) with entries in \(\mathbb{F}\), and with \(\operatorname{det}(A) \neq 0\), is a group with respect to matrix multiplication.

Show that two symmetric \(n \times n\) matrices \(A\) and \(B\) commute \((A B=B A)\) if and only if their product \(A B\) is symmetric.

A matrix \(\left(a_{i j}\right)\) is an upper-triangular matrix if \(a_{i j}=0\) whenever \(i>j\). Show that the space \(U\) of real \(n \times n\) upper-triangular matrices is a vector space. What is its dimension?

Let \(a, b\) and \(c\) be linearly independent vectors in \(\mathbb{R}^{3}\), and let \(\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be defined by \(\alpha(x)=(x \cdot a, x \cdot b, x \cdot c)\), where \(x \cdot y\) is the scalar product in \(\mathbb{R}^{3}\). What is \(\operatorname{det}(\alpha)\) ?

Let $$ A=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) $$ (i) Suppose that \(a d-b c \neq 0 .\) Find \(A^{-1}\). (ii) Suppose that \(a d-b c=0\). Find all column vectors \(x\) such that $$ \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right) $$

Show that if a \(2 \times 2\) complex matrix \(X\) commutes with every \(2 \times 2\) complex matrix, then \(X=\lambda I\) for some complex \(\lambda\), where \(I\) is the \(2 \times 2\) identity matrix.

Let \(\mathcal{E}=\\{(1,0),(0,1)\\}\) of \(\mathbb{C}^{2}\), and let $$ \alpha\left(z_{1}, z_{2}\right)=\left(3 z_{2}, i z_{1}\right), \quad \beta\left(z_{1}, z_{2}\right)=\left(z_{1}+2 i z_{2},(1+i) z_{1}\right) $$ Show that \(\alpha\) and \(\beta\) are linear maps of \(\mathbb{C}^{2}\) to itself. Find the matrix representations \(A\) of \(\alpha, B\) of \(\beta\), and \(C\) of \(\alpha \beta\) (in each case relative to \(\mathcal{E}\) ), and verify (by matrix multiplication) that \(C=A B\).

Let \(A=\left(\begin{array}{ll}1 & 2 \\ 3 & 6\end{array}\right)\), and for each \(2 \times 2\) real matrix \(X\) let \(\alpha(X)=A X\); thus \(\alpha\) is a map of the space \(M^{2 \times 2}(\mathbb{R})\) of real \(2 \times 2\) matrices into itself. Show that \(\alpha\) is a linear map. Find a basis of the kernel, and of the range, of \(\alpha\). [These bases should be made up of \(2 \times 2\) matrices, and the sum of the dimensions of these two subspaces should be four.]

Show that the space of \(n \times n\) matrices with trace zero is a vector space of dimension \(n^{2}-1\)

Let $$ A=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right), \quad B=\left(\begin{array}{ll} 0 & 2 \\ 3 & 3 \end{array}\right) $$ Show that \(B\) commutes with \(A(A B=B A)\). Show that the set of \(2 \times 2\) real matrices \(X\) that commute with \(A\) is a subspace \(M_{0}\) of the space of real \(2 \times 2\) matrices. Show also that \(\operatorname{dim}\left(M_{0}\right)=2\), and that \(I_{2}\) and \(B\) form a basis of \(M_{0} .\) As \(A\) and \(A^{2}\) commute with \(A\), this implies that \(A\) and \(A^{2}\) are linear combinations of \(I\) and \(B\). Find these linear combinations.

Let \(A\) be a real \(n \times n\) matrix. Show that the map \(\alpha: X \mapsto A X-X A\) is a linear map of \(M^{n \times n}(\mathbb{R})\) to itself. Show that the set of matrices \(X\) that commute with \(A\) is a subspace \(M(A)\) of \(M^{n \times n}(\mathbb{R})\). You should do this (a) by a direct argument, and (b) by considering the kernel of \(\alpha\). Is \(A\) in \(M(A)\) ? Now use a dimension argument to show that the map \(\alpha\) is not surjective, thus there is some matrix \(B\) such that \(B \neq A X-X A\) for any \(X\).