# Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular

## Problem 169

Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
$A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.$

Contents

## Hint.

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

## Solution.

Note that a matrix is singular if and only if its determinant is zero.
So we compute the determinant of the matrix $A$ as follows.
\begin{align*}
&\det(A)=\begin{vmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{vmatrix}\\
&=(-1)^{3+1}\cdot 0 \cdot \begin{vmatrix}
x^2 & 1\\
3& 1
\end{vmatrix}
+(-1)^{3+2}\cdot(-1)\cdot \begin{vmatrix}
x & 1\\
2& 1
\end{vmatrix}
+(-1)^{3+3}\cdot 1\cdot \begin{vmatrix}
x & x^2\\
2& 3
\end{vmatrix}\\
&\text{by the third row cofactor expansion}\\
&= 0+(x-2)+(3x-2x^2)\\
&=-2x^2+4x-2.
\end{align*}

Thus the determinant of $A$ is zero if
$\det(A)=-2x^2+4x-2=0,$ equivalently,
$x^2-2x+1=(x-1)^2=0.$ Thus, the determinant of the matrix $A$ is zero if and only if $x=1$.
Hence the matrix $A$ is singular if and only if $x=1$.

### More from my site

• 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 […]
• 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 […] • 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 […]
• 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 […]
• How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]
• 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 […]

#### You may also like...

##### 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...

Close