Find All Symmetric Matrices satisfying the Equation

Ohio State University exam problems and solutions in mathematics

Problem 697

Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix}
1 \\
-1
\end{bmatrix}
=
\begin{bmatrix}
2 \\
3
\end{bmatrix}$? Express your solution using free variable(s).

 
LoadingAdd to solve later

Sponsored Links


Solution.

Let $A=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$ be a $2\times 2$ matrix satisfying the conditions. Then as $A$ is symmetric, we have $A^{\trans}=A$. This yields that $b=c$.
So, we find all matrices $A=\begin{bmatrix}
a & b\\
b& d
\end{bmatrix}$ satisfying $A\begin{bmatrix}
1 \\
-1
\end{bmatrix}
=
\begin{bmatrix}
2 \\
3
\end{bmatrix}$.
We have
\begin{align*}
\begin{bmatrix}
2 \\
3
\end{bmatrix}=\begin{bmatrix}
a & b\\
b& d
\end{bmatrix}\begin{bmatrix}
1 \\
-1
\end{bmatrix}
=\begin{bmatrix}
a-b \\
b-d
\end{bmatrix}.
\end{align*}
Hence, we need $a-b=2$ and $b-d=3$.
Equivalently, $a=b+2, d=b-3$. So, we have
\begin{align*}
A=\begin{bmatrix}
a & b\\
b& d
\end{bmatrix}=\begin{bmatrix}
b+2 & b\\
b& b-3
\end{bmatrix}=b\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix}+\begin{bmatrix}
2 & 0\\
0& -3
\end{bmatrix},
\end{align*}
where $b$ is a free variable.

Common Mistake

This is a midterm exam problem of Lienar Algebra at the Ohio State University.

One common mistake is not using the assumption that $A$ is symmetric or using wrongly.
A matrix $A$ is symmetric if $A^{\trans}=A$. For a 2 by 2 matrix, this yields that the off-diagonal entries must be the same.
However, note that the diagonal entries can be distinct. Some students assumed the same diagonal entries and concluded that there are no matrices satisfying the conditions.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Linear Algebra Midterm 1 at the Ohio State University (3/3)Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […]
  • Maximize the Dimension of the Null Space of $A-aI$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 […]
  • Eigenvalues of a Hermitian Matrix are Real NumbersEigenvalues of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem)   We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […]
  • Using Properties of Inverse Matrices, Simplify the ExpressionUsing Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression \[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\] which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]
  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
  • If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero MatrixIf a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that \[AB=O,\] where $O$ is the $3\times 3$ zero matrix.   Proof. Since $A$ is singular, the equation $A\mathbf{x}=\mathbf{0}$ has a nonzero […]
  • The Product of Two Nonsingular Matrices is NonsingularThe Product of Two Nonsingular Matrices is Nonsingular Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem)   Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
  • 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 […]

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
Ohio State University exam problems and solutions in mathematics
Compute $A^5\mathbf{u}$ Using Linear Combination

Let \[A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix}...

Close