Conditions on Coefficients that a Matrix is Nonsingular

Linear Algebra Problems and Solutions

Problem 72

(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
\[ |b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}\] 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).

 

LoadingAdd to solve later

Sponsored Links


Hint.

  1. For (a), assume that the matrix $A$ is singular.
    Then the system of linear equation $A\mathbf{x}=\mathbf{0}$ has nonzero solution.
  2. For (b), apply (a).
  3. For(c), use part (a). Note  $e^2/2\pi^2=0.37433 \dots$.

Proof.

(a) Show that the matrix $A$ is nonsingular.

Suppose that the matrix $A$ is singular. Then the system of linear equations
\begin{align*}
a_{11}x_1+\cdots +a_{1n}x_n &=0 \\
\vdots \hspace{1.5cm}\vdots \hspace{0.5cm} &\hspace{0.5cm} \vdots \\
a_{n1}x_1+\cdots +a_{nn}x_n &=0 \\
\end{align*}
has nontrivial solution $(x_1,\dots, x_n) \neq (0, \dots, 0)$.
Since not all of $x_i$ are zero, let $x_k$ be the nonzero number such that $|x_k|$ is the largest among $|x_i|$.

That is,
\[|x_k|=\max_{1\leq i \leq n}{|x_i|}>0.\]


Then from the $k$-th equation in the system, we have
\[-a_{kk}x_k=a_{k1}x_1+\cdots+a_{k \, k-1}x_{k-1}+a_{k\, k+1}x_{k+1}+\cdots+a_{kn}x_n.\]

Taking the absolute values of both sides and by the triangle inequality, we have
\begin{align*}
|a_{kk}||x_k|&=|a_{k1}x_1+\cdots+a_{k \, k-1}x_{k-1}+a_{k\, k+1}x_{k+1}+\cdots+a_{kn}x_n|\\
& \leq |a_{k1}||x_1|+\cdots+|a_{k \, k-1}||x_{k-1}|+|a_{k\, k+1}||x_{k+1}|+\cdots+|a_{kn}||x_n|.
\end{align*}


Since $x_k$ is nonzero, we divide by $|x_k|$ and we have
\begin{align*}
|a_{kk}| & \leq |a_{k1}|\frac{|x_1|}{|x_k|}+\cdots+|a_{k \, k-1}|\frac{|x_{k-1}|}{|x_k|}+|a_{k\, k+1}|\frac{|x_{k+1}|}{|x_k|}+\cdots+|a_{kn}|\frac{|x_n|}{|x_k|}.
\end{align*}
By the choice of $x_k$, the absolute value of $x_k$ is maximal, hence $|x_i|/|x_k|\leq 1$ for all $i$.

Therefore we have
\[|a_{kk}| \leq |a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|.\] However this contradicts the given condition. Therefore the matrix $A$ is nonsingular.

(b) Prove that the matrix $B$ is nonsingular.

Note that
\begin{align*}
&|b_{i1}|+\cdots +|b_{i\,i-1}|+|b_{i \, i+1}|+\cdots +|b_{in}|\\
& <\frac{1}{n-1}+\cdots+ \frac{1}{n-1}=\frac{n-1}{n-1}=1=|b_{ii}|.
\end{align*}
Therefore the matrix $B$ satisfy the condition of part (a). Therefore from the result of (a), the matrix $B$ is nonsingular.

(c) Determine whether the matrix is nonsingular or not.

Let us apply part (a) to the matrix $C$.
We have
\[\frac{e^2}{2\pi^2}=0.37433 \dots. \] Thus we calculate
\[e+\frac{e^2}{2\pi^2}=3.09262\dots <\pi=3.14159\dots\]

Thus the sum of the (absolute value of) off-diagonal entries are less than the diagonal entry. Hence by part (a), the matrix $C$ is nonsingular.

Comment.

These are simple and interesting criteria for the non singularity of a matrix.
You may easily cook up matrices satisfying (or not satisfying) the conditions.
Why don’t you make one and give it to your friend for a math quiz?


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

2 Responses

  1. Pavel says:

    Hello.

    I believe that in the proof “Suppose that the matrix A is [non]singular. Then the system of linear equations …
    has nontrivial solution …”. must be replaced with “Suppose that the matrix A is singular…”

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear Transformation problems and solutions
Matrix Representations for Linear Transformations of the Vector Space of Polynomials

Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less....

Close