As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$.

Let us think backwards. Suppose that we have $MN=P$ for some matrix $N$, which we want to find.
Then multiply $M^{-1}$ on the left of the equation $MN = P$ yields $N = M^{-1} P$.

This is the matrix we are looking for, as $ M N = M ( M^{-1} P) = P$.

If a Matrix is the Product of Two Matrices, is it Invertible?
(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
\[A=BC,\]
where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.
Prove that the matrix $A$ cannot be invertible.
(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be […]

A Matrix is Invertible If and Only If It is Nonsingular
In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:
A matrix $A$ is nonsingular if and only if $A$ is invertible.
(a) Show that if $A$ is invertible, then $A$ is […]

Nilpotent Matrices and Non-Singularity of Such Matrices
Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.
Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]

Find a Nonsingular Matrix Satisfying Some Relation
Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\]
where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
1 & 2 & […]

Two Matrices are Nonsingular if and only if the Product is Nonsingular
An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.
(a) If $A$ and $B$ are $n\times […]

Find the Inverse Matrix of a $3\times 3$ Matrix if Exists
Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 1 & 2 \\
0 &0 &1 \\
1 & 0 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(The Ohio State University, Linear Algebra Midterm Exam […]

Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$
(a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix}
2 & 0 & -1 \\
0 &2 &-1 \\
-1 & 0 & 1
\end{bmatrix}.\]
(b) Find the inverse matrix of $A$.
Solution
(a) Find a $3\times 3$ nonsingular matrix $A$.
Assume […]

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 $B$ […]