An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that

$AB=I$, and

$BA=I$,

where $I$ is the $n\times n$ identity matrix.

If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted by $A^{-1}$.

In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.
So if we know $AB=I$, then we can conclude that $B=A^{-1}$.

Let $A$ and $B$ be $n\times n$ matrices.
Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.

Since $AB=I$, we have
\begin{align*}
\det(A)\det(B)=\det(AB)=\det(I)=1.
\end{align*}
This implies that the determinants $\det(A)$ and $\det(B)$ are not zero.
Hence $A, B$ are invertible matrices: $A^{-1}, B^{-1}$ exist.

Now we compute
\begin{align*}
I&=BB^{-1}=BIB^{-1}\\
&=B(AB)B^{-1} &&\text{since $AB=I$}\\
&=BAI=BA.
\end{align*}
Hence we obtain $BA=I$.
Since $AB=I$ and $BA=I$, we conclude that $B=A^{-1}$.

Find 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 […]

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 […]

Quiz 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 […]

Problems and Solutions About Similar Matrices
Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.
(a) If $A$ is similar to $B$, then $B$ is similar to $A$.
(b) $A$ is similar to itself.
(c) If $A$ is similar to $B$ and $B$ […]

Sherman-Woodbery Formula for the Inverse Matrix
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\]
Define the matrix […]

The Inverse Matrix is Unique
Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.
Hint.
That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$.
(That's why we say "the" inverse matrix of $A$ and denote it by […]

Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite
Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
Proof.
(a) Prove that $A$ is […]

Invertible Idempotent Matrix is the Identity Matrix
A square matrix $A$ is called idempotent if $A^2=A$.
Show that a square invertible idempotent matrix is the identity matrix.
Proof.
Let $A$ be an $n \times n$ invertible idempotent matrix.
Since $A$ is invertible, the inverse matrix $A^{-1}$ of $A$ exists and it […]