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.)

Determine 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.
[…]

Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular
Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
\[A=\begin{bmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{bmatrix}.\]
Hint.
Use the fact that a matrix is singular if and only if its determinant is […]

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 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 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 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 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 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 […]