Rank and Determinants

Let $M$ be a matrix with entries in a field $\mathbf{F}$. The row rank of $M$ over $\mathbf{F}$ is the maximal number of rows which are linearly independent (as vectors) over $\mathbf{F}$. The column rank is similarly defined using columns instead of rows.

Let the $n \times n$ real matrix $A$ be diagonalizable and have a one-dimensional null space. Prove that a nonzero left null vector of $A$ cannot be orthogonal to a nonzero right null vector of $A$.

Let $A$ be an $n \times n$ real matrix and $A^{t}$ its transpose. Show that $A^{t}A$ and $A^{t}$ have the same range.

Suppose that $P$ and $Q$ are $n \times n$ matrices such that $P^{2} = P$, $Q^{2} = Q$, and $1-(P+Q)$ is invertible. Show that $P$ and $Q$ have the same rank.

Let $A(m,n)$ be the $m \times n$ matrix with entries

where $0^{0} = 1$ by definition. Regarding the entries of $A(m,n)$ as representing congruence classes (mod $p$), determine the rank of $A(m,n)$ over the finite field $\mathbf{F}_{p} = \mathbb {Z}^{p}$ for all $m, n \geq 1$ and all primes $p$.

Let $A$ be an $n \times n$ matrix over a field $K$. Prove that

Let $T$ be a real, symmetric, $n \times n$, tridiagonal matrix:

(All entries not on the main diagonal or the diagonals just above and below the main one are zero.) Assume $b_{j} \neq 0$ for all $j$.

Prove:

Let $A = (a_{ij})$ be an $n \times n$ real matrix satisfying the conditions:

Show that $\operatorname {det}\left(A\right) > 0$.

Let $A = (a_{ij})_{i,j=1}^{r}$ be a square matrix with integer entries.

Let $A$ be a symmetric $n \times n$ matrix over $\mathbb {R}$ of rank $n-1$. Prove there is a $k$ in $\left\{ 1,2,\ldots ,n\right\}$ such that the matrix resulting from deletion of the $k$-th row and $k$-th column from $A$ has rank $n-1$.

Let $\mathbb {R}^{x_{1}, \ldots , x_{n}}$ be the polynomial ring over the real field $\mathbb {R}$ in the $n$ variables $x_{1}, \ldots , x_{n}$. Let the matrix $A$ be the $n \times n$ matrix whose $i$-th row is $(1, x_{i}, x_{i}^{2}, \ldots , x_{i}^{n-1})$, $i = 1, \ldots , n$. Show that

A matrix of the form

where the $a_{i}$ are complex numbers, is called a Vandermonde matrix.

Let $A$ and $B$ be $n \times n$ complex unitary matrices. Prove that $\left|\det (A+B)\right| \leq 2^{n}$.

Give an example of a continuous function $v : \mathbb {R} \to \mathbb {R}^{3}$ with the property that $v(t_{1})$, $v(t_{2})$, and $v(t_{3})$ form a basis for $\mathbb {R}^{3}$ whenever $t_{1}, t_{2}$, and $t_{3}$ are distinct points of $\mathbb {R}$.

Let $f_{1}, f_{2}, \ldots , f_{n}$ be continuous real valued functions on $[a,b]$. Show that the set $\left\{ f_{1}, \ldots , f_{n}\right\}$ is linearly dependent on $[a,b]$ if and only if

Let $M_{2 \times 2}$ be the vector space of all real $2 \times 2$ matrices. Let

and define a linear transformation $L : M_{2 \times 2} \to M_{2 \times 2}$ by $L(X) = AXB$. Compute the trace and the determinant of $L$.

Let $V$ be the vector space of all real $3 \times 3$ matrices and let $A$ be the diagonal matrix

Calculate the determinant of the linear transformation $T$ on $V$ defined by $T(X) = \frac{1}{2}(AX+XA)$.

Let $M_{3 \times 3}$ denote the vector space of real $3 \times 3$ matrices. For any matrix $A \in M_{3 \times 3}$, define the linear operator $L_{A} : M_{3 \times 3} \to M_{3 \times 3}$, $L_{A}(B) = AB$. Suppose that the determinant of $A$ is $32$ and the minimal polynomial is $(t-4)(t-2)$. What is the trace of $L_{A}$?

Let $S$ denote the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_{A} : S \to S$, $T_{A}(X) = AXA^{t}$.

Let $M_{7 \times 7}$ denote the vector space of real $7 \times 7$ matrices. Let $A$ be a diagonal matrix in $M_{7 \times 7}$ that has $+1$ in four diagonal positions and $-1$ in three diagonal positions. Define the linear transformation $T$ on $M_{7 \times 7}$ by $T(X) = AX-XA$. What is the dimension of the range of $T$?

Let $\mathbf{F}$ be a field. For $m$ and $n$ positive integers, let $M_{m \times n}$ be the vector space of $m \times n$ matrices over $\mathbf{F}$. Fix $m$ and $n$, and fix matrices $A$ and $B$ in $M_{m \times n}$. Define the linear transformation $T$ from $M_{n \times m}$ to $M_{m \times n}$ by

Prove that if $m \neq n$, then $T$ is not invertible.