We claim that the statement is false.
As a counterexample, consider the matrices
\[A=\begin{bmatrix}
1 & 0\\
0& 0
\end{bmatrix} \text{ and } B=\begin{bmatrix}
0 & 0\\
0& 1
\end{bmatrix}.\]
Then
\[A+B=\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}\]
and we have
\[\det(A+B)=\begin{vmatrix}
1 & 0\\
0& 1
\end{vmatrix}=1.\]
On the other hand, the determinants of $A$ and $B$ are
\[\det(A)=0 \text{ and } \det(B)=0,\]
and hence
\[\det(A)+\det(B)=0\neq 1=\det(A+B).\]
Therefore, the statement is false and in general we have
\[\det(A+B)\neq \det(A)+\det(B).\]
Remark.
When we computed the $2\times 2$ matrices, we used the formula
\[\begin{vmatrix}
a & b\\
c& d
\end{vmatrix}=ad-bc.\]
This problem showed that the determinant does not preserve the addition.
However, the determinant is multiplicative.
In general, the following is true:
\[\det(AB)=\det(A)\det(B).\]
12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]
If the Matrix Product $AB=0$, then is $BA=0$ as Well?
Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.
Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a […]
True or False. Every Diagonalizable Matrix is Invertible
Is every diagonalizable matrix invertible?
Solution.
The answer is No.
Counterexample
We give a counterexample. Consider the $2\times 2$ zero matrix.
The zero matrix is a diagonal matrix, and thus it is diagonalizable.
However, the zero matrix is not […]
True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces
Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample.
Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.
If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a […]
Is there an Odd Matrix Whose Square is $-I$?
Let $n$ be an odd positive integer.
Determine whether there exists an $n \times n$ real matrix $A$ such that
\[A^2+I=O,\]
where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix.
If such a matrix $A$ exists, find an example. If not, prove that […]
Similar Matrices Have the Same Eigenvalues
Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
Proof.
We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]