Linear Algebra

Find All Values of $x$ so that a Matrix is Singular

Problem 168

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.

Add to solve later

Hint.

Use the fact that a matrix is singular if and only if the determinant of the matrix is zero.

To compute the determinant, use a cofactor expansion.

Solution.

We use the fact that a matrix is singular if and only if the determinant of the matrix is zero.

We compute the determinant of $A$ as follows.
\begin{align*}
&\det(A)= \begin{vmatrix}
1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{vmatrix}\\[6pt] &=\begin{vmatrix}
1 & -x & 0 \\
0 &1 &-x \\
1 & 0 & -1
\end{vmatrix}
-0\begin{vmatrix}
-x & 0 & 0 \\
0 &1 &-x \\
1 & 0 & -1
\end{vmatrix}+0\begin{vmatrix}
-x & 0 & 0 \\
1 &-x &0 \\
1 & 0 & -1
\end{vmatrix}+0\begin{vmatrix}
-x & 0 & 0 \\
1 &-x &0 \\
0 & 1 & -x
\end{vmatrix}
\\[6pt] & \text{ by the first column cofactor expansion}\\[6pt] &=\begin{vmatrix}
1 & -x & 0 \\
0 &1 &-x \\
1 & 0 & -1
\end{vmatrix}\\[6pt] &=\begin{vmatrix}
1 & -x\\
0& -1
\end{vmatrix}-0\begin{vmatrix}
-x & 0\\
0& -1
\end{vmatrix}+\begin{vmatrix}
-x & 0\\
1& -x
\end{vmatrix}\\[6pt] & \text{ by the first column cofactor expansion}\\[6pt] &=-1+x^2.
\end{align*}

Therefore we have $\det(A)=x^2-1$. Thus $\det(A)=0$ if and only if $x=\pm 1$.
We conclude that the matrix $A$ is singular if and only if $x=\pm 1$.

Comment.

You may use the rule of Sarrus to compute the $3\times 3$ determinant instead of the cofactor expansion if you like so.

Add to solve later

More from my site

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 Values of $h$ so that the Given Vectors are Linearly Independent Find the value(s) of $h$ for which the following set of vectors \[\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […]
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 […]
$Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular$ 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 […]
How to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix Find the inverse matrix of the $3\times 3$ matrix \[A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}\] using the Cayley-Hamilton theorem. Solution. To apply the Cayley-Hamilton theorem, we first determine the characteristic […]
Properties of Nonsingular and Singular Matrices 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 […]
Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to 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 […]