Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

Ohio State University exam problems and solutions in mathematics

Problem 486

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 matrix $A^{-1}$.

(The Ohio State University, Linear Algebra Final Exam Problem)
 
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Solution.

Assume that $A$ is nonsingular.
Then the inverse matrix $A^{-1}$ exists and we have
\begin{align*}
A&=A^{-1}A^4A^{-2}\\
&=A^{-1}(ABA^2+2A^3)A^{-2}\\
&=A^{-1}ABA^2 A^{-2}+2A^{-1}A^3A^{-2}\\
&=B+2I,
\end{align*}
where $I$ is the $3\times 3$ identity matrix.

Thus we have
\begin{align*}
A=B+2I=\begin{bmatrix}
1 & 1 & -1 \\
0 &1 &0 \\
2 & 1 & -2
\end{bmatrix}.
\end{align*}

We compute the determinant of $A$ by the second row cofactor expansion as follows.
\begin{align*}
\det(A)=\begin{vmatrix}
1 & 1 & -1 \\
0 &1 &0 \\
2 & 1 & -2
\end{vmatrix}
=\begin{vmatrix}
1 & -1\\
2& -2
\end{vmatrix}=0.
\end{align*}
This contradicts the assumption that $A$ is nonsingular.

Therefore, there is no nonsingular matrix $A$ satisfying the relation $A^4=ABA^2+2A^3$.

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

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