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 later Tagged: invertible
Add to solve later If a Matrix is the Product of Two Matrices, is it Invertible?
Problem 393
(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 later If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?
Problem 317
Suppose 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)
Add to solve later Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal
Problem 172
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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