True or False Problems on Midterm Exam 1 at OSU Spring 2018

Ohio State University exam problems and solutions in mathematics

Problem 702

The following problems are True or False.

Let $A$ and $B$ be $n\times n$ matrices.

(a) If $AB=B$, then $B$ is the identity matrix.
(b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions.
(c) If $A$ is invertible, then $ABA^{-1}=B$.
(d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix.
(e) If $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions.

 
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Solution.

(a) True or False: if $AB=B$, then $B$ is the identity matrix.

False. For example, if $B$ is the zero matrix, then of course we have $AB=B$ as both sides are the zero matrix.

(b) True or False: if the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions.

False. If the coefficient matrix $A$ is invertible, the system has a unique solution $\mathbf{x}=A^{-1}\mathbf{b}$.

(c) True or False: if $A$ is invertible, then $ABA^{-1}=B$.

False. The given equality is equivalent to $AB=BA$. Even $A$ is invertible, matrix multiplication is not commutative. As a counterexample, consider
\[A=\begin{bmatrix}
1 & 1\\
0& 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
1 & 0\\
-1& 1
\end{bmatrix}.\] Note that the determinant of $A$ is $\det(A)=1\neq 0$. Hence $A$ is invertible.
Yet, we have
\[AB=\begin{bmatrix}
1 & 1\\
0& 1
\end{bmatrix}\begin{bmatrix}
1 & 0\\
-1& 1
\end{bmatrix}=\begin{bmatrix}
0 & 1\\
-1& 1
\end{bmatrix}\] and
\[BA=\begin{bmatrix}
1 & 0\\
-1& 1
\end{bmatrix}\begin{bmatrix}
1 & 1\\
0& 1
\end{bmatrix}=\begin{bmatrix}
1 & 1\\
-1& 0
\end{bmatrix},\] and hence $AB\neq BA$.

(d) True or False: if $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix.

True. Since $A$ is idempotent, we have $A^2=A$. As $A$ is nonsingular, it is invertible. Thus, the inverse matrix $A^{-1}$ exists. Then we have
\[I=A^{-1}A=A^{-1}A^2=IA=A.\] Hence, such a matrix $A$ must be the identity matrix $I$.

(e) True or False: if $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions.

Note that any homogeneous system has the zero solution. In addition to the zero solution, this system has a solution $x_1=0, x_2=0, x_3=1$. So, it has at least two solutions. The only possibilities for the number of solutions of a system of linear equations are zero, one, or infinitely many.
So, we conclude that the system must have infinitely many solutions.


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