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

Problems and solutions in Linear Algebra

Problem 169

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

 
LoadingAdd to solve later

Sponsored Links


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 $x=1$.


LoadingAdd to solve later

Sponsored Links

More from my site

  • 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 […]
  • 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 […]
  • A Relation of Nonzero Row Vectors and Column VectorsA Relation of Nonzero Row Vectors and Column Vectors Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that \[\mathbf{y}A=\mathbf{y}.\] (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that \[A\mathbf{x}=\mathbf{x}.\] (Here a […]
  • 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 […]
  • Find Values of $h$ so that the Given Vectors are Linearly IndependentFind 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 […]
  • Properties of Nonsingular and Singular MatricesProperties 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 […]
  • If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is SingularIf the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero. Then prove that the matrix $A$ is singular.   Definition. An $n\times n$ matrix $A$ is said to be singular if there exists a nonzero vector $\mathbf{v}$ such that […]
  • Determine Conditions on Scalars so that the Set of Vectors is Linearly DependentDetermine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear algebra problems and solutions
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