## If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$

## Problem 657

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

Add to solve laterSuppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

Add to solve later**(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 written as

\[A=BC,\]
where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

Add to solve laterSuppose that $A$ is a real $n\times n$ matrix.

**(a)** Is it true that $A$ must commute with its transpose?

**(b)** Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.

Is it true that the rows of $A$ must also form an orthonormal set?

(*University of California, Berkeley, Linear Algebra Qualifying Exam*)

Let $R$ be a commutative ring.

Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.

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