How to Prove a Matrix is Nonsingular in 10 Seconds

How to Prove a Matrix is Nonsingular in 10 Seconds!!

Problem 509

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.

 
LoadingAdd to solve later

Sponsored Links


Proof.

To show that the matrix $A$ is nonsingular, it suffices to prove that $\det(A)\neq 0$.

One way is to compute the determinant of $A$ directly.

However, as the numbers in $A$ are quite large for hand computation, the direct calculation must be tedious.


So we consider an alternative method.
Note that we do not have to find the exact value of $\det(A)$, but we just need to know $\det(A)\neq 0$.

Thus, it suffices to show that $\det(A)$ is odd. ($0$ is an even number.)
This suggests that considering the matrix modulo $2$ is helpful.


Let $\bar{A}$ be the matrix whose $(i, j)$-entry is the $(i,j)$-entry of $A$ modulo $2$.
That is,
\begin{align*}
\bar{A}:=\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
\end{align*}
(Remark that the diagonal entries of $A$ are odd, and off-diagonal entries are even.)

Since $\det(A)$ is a polynomial of entries of $A$, we have
\begin{align*}
\det(A) &\equiv \det(\bar{A}) \pmod{2}\\
&=1.
\end{align*}

It follows that $\det(A)$ is odd, and in particular $\det(A)\neq 0$.
Thus the matrix $A$ is nonsingular.

What’s $\det(A)$ anyway?

Just for the record, the determinant of $A$ is
\[\det(A)=-20330769121541702776233175.\]

Beautiful Formulas for $\pi$

This problem was nothing to do with the number $\pi$ (except we used the digits of $\pi$) and the matrix is far from beautiful.
(Although the method we used is beautiful.)

Check out the post
Beautiful formulas for pi=3.14…
for beautiful formulas containing $\pi$ like

\[\pi= \cfrac{4}{1
+ \cfrac{1^2}{2
+ \cfrac{3^2}{2 + \cfrac{5^2}{2
+\cfrac{7^2}{2 + \cdots}}}}}\]

LoadingAdd to solve later

Sponsored Links

More from my site

  • Determine whether the Matrix is Nonsingular from the Given RelationDetermine whether the Matrix is Nonsingular from the Given Relation Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$. If \[A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},\] then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. […]
  • If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$ Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.   Proof. As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$. Let us think backwards. Suppose that […]
  • Nilpotent Matrices and Non-Singularity of Such MatricesNilpotent Matrices and Non-Singularity of Such Matrices Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix. Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]
  • Compute Determinant of a Matrix Using Linearly Independent VectorsCompute Determinant of a Matrix Using Linearly Independent Vectors Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have \[A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […]
  • Find All Values of $x$ so that a Matrix is SingularFind All Values of $x$ so that a Matrix is Singular Let \[A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}\] be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.   Hint. Use the fact that a matrix is singular if and only […]
  • A Matrix is Invertible If and Only If It is NonsingularA Matrix is Invertible If and Only If It is Nonsingular In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility. That is, we will prove that: A matrix $A$ is nonsingular if and only if $A$ is invertible. (a) Show that if $A$ is invertible, then $A$ is […]
  • Determine whether the Given 3 by 3 Matrices are NonsingularDetermine whether the Given 3 by 3 Matrices are Nonsingular 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) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$.   Solution. Recall that […]
  • Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]

You may also like...

More in Linear Algebra
Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra
Eigenvalues of a Matrix and its Transpose are the Same

Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues...

Close