Nonsingular Matrices

Nonsingular Matrices

Definition

Let $A$ be an $n\times n$ matrix.

  1. $A$ is nonsingular if the only solution to $A\mathbf{x}=\mathbf{0}$ is the zero solution $\mathbf{x}=\mathbf{0}$.
Summary

Let $A$ be an $n\times n$ matrix.

  1. If $A$ is nonsingular, then $A^{\trans}$ is nonsingular.
  2. $A$ is nonsingular if and only if the column vectors of $A$ are linearly independent.

=solution

Problems

  1. Let $A$ be the following $3 \times 3$ matrix.
    \[A=\begin{bmatrix}
    1 & 1 & -1 \\
    0 &1 &2 \\
    1 & 1 & a
    \end{bmatrix}.\] Determine the values of $a$ so that the matrix $A$ is nonsingular.

  2. Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that
    \[AB=O,\] where $O$ is the $3\times 3$ zero matrix.

  3. Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that $AB=O$, where $O$ is the $n\times n$ zero matrix.

  4. Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.
    (a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.
    (b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?
    (c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.

  5. Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero. Then prove that the matrix $A$ is singular.
  6. Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.
    (The Ohio State University)

  7. (a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.
    \[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\] for all $1 \leq i \leq n$. Show that the matrix $A$ is nonsingular.
    (b) Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation
    \[ |b_{i\,i}|=1 \text{ and } |b_{ij}|<\frac{1}{n-1}\] for all $i$ and $j$ with $i \neq j$. Prove that the matrix $B$ is nonsingular.
    (c) Determine whether the following matrix is nonsingular or not.
    \[C=\begin{bmatrix}
    \pi & e & e^2/2\pi^2 \\[5 pt] e^2/2\pi^2 &\pi &e \\[5pt] e & e^2/2\pi^2 & \pi
    \end{bmatrix},\] where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).

  8. Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}$. (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{x}$. (Here a column vector means an $n \times 1$ matrix.)
  9. Without using properties of determinants
    Determine whether each of the following statement is True or False.
    (a) Suppose that $A$ and $B$ are nonsingular $n\times n$ matrices. Then $A+B$ is nonsingular.
    (b) If a square matrix has no zero rows or columns, then it has an inverse matrix.
    (c) Let $A$ be an $m \times n$ matrix.
    If the equation $A\mathbf{x}=\mathbf{0}$ has only the trivial solution $\mathbf{x}\in \R^n$, then the columns of $A$ are linearly independent.
    (d) Let $A$ be an $m \times n$ matrix.
    If the equation $A\mathbf{x}=\mathbf{0}$ has only the trivial solution $\mathbf{x}\in \R^n$, then the rows of $A$ are linearly independent.
    (e) The row echelon form of an $3\times 3$ matrix is invertible.
    (f) There is a non-zero nonsingular matrix $A$ such that $A^2=O$.
    (g) If $A$ and $B$ are invertible $n\times n$ matrices, then $AB=BA$.
    (h) If $A$ and $B$ are $n\times n$ nonsingular matrices such that $A^2=I$ and $B^2=I$, then $(AB)^{-1}=BA$.
    (i) If $A$ is an $m \times n$ matrix such that $A\mathbf{x}=\mathbf{0}$ for every vector $\mathbf{x}$ in $\R^n$, then $A$ is the $m\times n$ zero matrix.
    (j) Let $A$ be a $2 \times 2$ nonsingular matrix and let $\mathbf{v}_1$ and $\mathbf{v}_2$ be linearly independent vectors in $\R^2$.
    Then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are linearly independent vectors in $\R^2$.

  10. Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.
  11. Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector. Then the product $A\mathbf{b}$ is an $n$-dimensional vector. Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$. Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.
  12. An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements.
    (a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.
    (b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:
    1. The matrix $B$ is nonsingular.
    2. The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)

  13. Let $A$ be a singular $n\times n$ matrix.
    Let
    \[\mathbf{e}_1=\begin{bmatrix}
    1 \\
    0 \\
    \vdots \\
    0
    \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
    0 \\
    1 \\
    \vdots \\
    0
    \end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
    0 \\
    0 \\
    \vdots \\
    1
    \end{bmatrix}\] be unit vectors in $\R^n$.

    Prove that at least one of the following matrix equations
    \[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.

  14. Using the numbers appearing in
    \[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix}
    3 & 14 &1592& 65358\\
    97932& 38462643& 38& 32\\
    7950& 2& 8841& 9716\\
    939937510& 5820& 974& 9
    \end{bmatrix}.\] Prove that the matrix $A$ is nonsingular.