Properties of Nonsingular and Singular Matrices

Linear algebra problems and solutions

Problem 25

An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$.
Otherwise $A$ is called singular.

(a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

(b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.

(c) Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$.

Restriction
Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible.

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Proof.

(a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

Consider the equation $AB\mathbf{x}=\mathbf{0}$
Let $\mathbf{y}=B\mathbf{x}$. Then we have $A\mathbf{y}=\mathbf{0}$.

Then $\mathbf{y}$ must be the zero vector since $A$ is nonsingular.
Thus we have $B\mathbf{x}=\mathbf{0}$.

Since $B$ is nonsingular, we must have $\mathbf{x}=\mathbf{0}$.
Hence the only solution to $AB\mathbf{x}=\mathbf{0}$ is the zero vector, and we conclude that $AB$ is nonsingular.

(b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.

Suppose that $A$ is an $n\times n$ nonsingular matrix. Let $A_i$ be the $i$-th column vector of $A$ for $i=1,2,\dots, n$.

Suppose that we have a linear combination
\[x_1 A_1+x_2 A_2+\cdots +x_n A_n=\mathbf{0} \tag{*}.\] This equation can be written as
\[ A\mathbf{x}=\mathbf{0},\] where $A=[A_1A_2\dots A_n]$ and $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
\vdots\\
x_n
\end{bmatrix}$.

Since $A$ is nonsingular, we must have $\mathbf{x}=\mathbf{0}$, hence $x_1=x_2=\cdots =x_n=0$.
Thus only solution to (*) is zero coefficients. Thus the vectors $A_i$ are linearly independent.

(c) Show that $A$ is nonsingular if and only if $A\mathbf{x}=\mathbf{b}$ has a unique solution for any $\mathbf{b}\in \R^n$.

$(\Leftarrow)$

If $A\mathbf{x}=\mathbf{b}$ has a unique solution for any $\mathbf{b}$, then choose $\mathbf{b}=\mathbf{0}$.
Then $A\mathbf{x}=\mathbf{0}$ has a unique solution.

Note that this equation has the trivial solution $\mathbf{x}=\mathbf{0}$.
Thus $\mathbf{x}=\mathbf{0}$ is the unique solution of $A\mathbf{x}=\mathbf{0}$, hence $A$ is nonsingular.

$(\Rightarrow)$

Assume that $A$ is nonsingular. Let $\mathbf{b}$ be a vector in $\R^n$ and consider the equation $A\mathbf{x}=\mathbf{b}$.
We prove that

  1. the existence of a solution, and
  2. the uniqueness of the solution.

1. The existence of a solution

We show that $A\mathbf{x}=\mathbf{b}$ has at least one solution .
Let us write $A=[A_1A_2\dots A_n]$, where $A_i$ is the $i$-th column vector of the matrix $A$.

Then the set $\{A_1, A_2, \dots, A_n, \mathbf{b} \}$ contains $n+1$ vectors in $R^n$. Thus the set must be linearly dependent.

So there exist $c_i$, $i=1,2, \dots, n+1$, such that not all of $c_i$ are zero and we have
\[c_1A_1+c_2 A_2+\cdots c_n A_n+c_{n+1} \mathbf{b}=\mathbf{0} \tag{**}.\] We claim that $c_{n+1} \neq 0$. If $c_{n+1}=0$, then we have
\[c_1A_1+c_2 A_2+\cdots c_n A_n=\mathbf{0}\] and by part (b), the vectors $A_i$ are linearly independent, hence $c_i=0$ for all $i$.

This contradicts the choice of $c_i$. Thus we must have $c_{n+1}\neq 0$.
Since $c_{n+1}\neq 0$, we divide (**) by $c_{n+1}$ and solve for $\mathbf{b}$, and obtain
\[ \mathbf{b}=-\frac{c_1}{c_{n+1}} A_1 – \cdots -\frac{c_n}{c_{n+1}}A_n.\] Therefore the equation $A\mathbf{x}=\mathbf{b}$ has solution
\[x_i=-\frac{c_i}{c_{n+1}}.\]

2. The uniqueness of the solution

Now we prove that $A\mathbf{x}=\mathbf{b}$ has a unique solution.
Suppose that $\mathbf{x}_1$ and $\mathbf{x}_2$ are solutions.

Thus we have
\[A \mathbf{x}_1=\mathbf{b}=A\mathbf{x}_2.\] Then we have $A(\mathbf{x}_1-\mathbf{x}_2)=\mathbf{0}$.

Since $A$ is nonsingular, we must have $\mathbf{x}_1-\mathbf{x}_2=\mathbf{0}$, hence $\mathbf{x}_1=\mathbf{x}_2$.
This proves the uniqueness of the solution.

Related Question.

The converse statment of (a) is also true:

Let $A, B$ be $n\times n$ matrices and suppose $AB$ is nonsingular. Then $A$ and $B$ are nonsingular.

See the post ↴
Two Matrices are Nonsingular if and only if the Product is Nonsingular


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1 Response

  1. 07/29/2016

    […] basic properties of a nonsingular matrix, see the problem Properties of nonsingular and singular matrices. The result of this problem will be used in the proof […]

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