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.
3. $A\mathbf{x}=\mathbf{b}$ has a unique solution for every $n\times 1$ column vector $\mathbf{b}$ if and only if $A$ is nonsingular.

=solution

Problems

1. Determine whether the following matrices are nonsingular or not.
(a) $A=\begin{bmatrix} 1 & 0 & 1 \\ 2 &1 &2 \\ 1 & 0 & -1 \end{bmatrix}$.
(b) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$.

2. Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.
(a) Show that $M$ is singular.
(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.

3. 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.

4. Determine the values of a real number $a$ such that the matrix $A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}$ is nonsingular.

5. (a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?
(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular?
(c) Let $A$ be a $4\times 4$ matrix and let $\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 4 \\ 3 \\ 2 \\ 1 \end{bmatrix}$. Suppose that we have $A\mathbf{v}=A\mathbf{w}$. Is the matrix $A$ nonsingular?

6. 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.

7. 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.

8. 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.

9. 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.
10. 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)

11. (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).

12. 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.)
13. 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 $\R^2$.

14. Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.
15. 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 $\mathbf{b}$.
16. 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.)

17. 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 $\mathbf{x}\in \R^n$.

18. 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.