Find All Values of $x$ such that the Matrix is Invertible

Inverse Matrices Problems and Solutions

Problem 721

Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if
\[
A=
\begin{bmatrix}
1 & 0 & c \\
0 & a & -b \\
-1/a & x & x^{2}
\end{bmatrix}
.
\]

 
LoadingAdd to solve later

Solution.

We know that $A$ is invertible precisely when $\det(A)\neq 0$. We therefore compute, by expanding along the first row,
\begin{align*}
\det(A)
&=
1
\begin{vmatrix}
a & -b \\ x & x^{2}
\end{vmatrix}
+c
\begin{vmatrix}
0 & a \\ -1/a & x
\end{vmatrix}
=
1(ax^{2}+bx)
+c(0+1)
\\
&=
ax^{2}+bx+c
.
\end{align*}
Thus $\det(A)\neq 0$ when $ax^{2}+bx+c\neq 0$. We know by the quadratic formula that $ax^{2}+bx+c=0$ precisely when
\[
x=
\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2}
.
\] Therefore, $A$ is invertible so long as $x$ satisfies both of the following inequalities:
\[
x\neq
\dfrac{-b+\sqrt{b^{2}-4ac}}{2}
,\quad
x\neq
\dfrac{-b-\sqrt{b^{2}-4ac}}{2}
.
\]


LoadingAdd to solve later

More from my site

  • Find Inverse Matrices Using Adjoint MatricesFind Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
  • For Which Choices of $x$ is the Given Matrix Invertible?For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix \[A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}\] is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]
  • Find All the Eigenvalues of Power of Matrix and Inverse MatrixFind All the Eigenvalues of Power of Matrix and Inverse Matrix Let \[A=\begin{bmatrix} 3 & -12 & 4 \\ -1 &0 &-2 \\ -1 & 5 & -1 \end{bmatrix}.\] Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.   Proof. We first determine all the eigenvalues of the matrix […]
  • Eigenvalues and their Algebraic Multiplicities of a Matrix with a VariableEigenvalues and their Algebraic Multiplicities of a Matrix with a Variable Determine all eigenvalues and their algebraic multiplicities of the matrix \[A=\begin{bmatrix} 1 & a & 1 \\ a &1 &a \\ 1 & a & 1 \end{bmatrix},\] where $a$ is a real number.   Proof. To find eigenvalues we first compute the characteristic polynomial of the […]
  • Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a RelationQuiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of \[A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that \[A^3=A^2B-3A^2,\] where […]
  • Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ Determine whether there exists a nonsingular matrix $A$ if \[A^4=ABA^2+2A^3,\] where $B$ is the following matrix. \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\] If such a nonsingular matrix $A$ exists, find the inverse […]
  • Compute the Determinant of a Magic SquareCompute the Determinant of a Magic Square Let \[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic […]
  • The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate […]

You may also like...

Leave a Reply

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

More in Linear Algebra
Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra
Find All Eigenvalues and Corresponding Eigenvectors for the $3\times 3$ matrix

Find all eigenvalues and corresponding eigenvectors for the matrix $A$ if \[ A= \begin{bmatrix} 2 & -3 & 0 \\...

Close