Equalizers and coequalizers of matrices

Suppose $F$ is a field and $A, B$ are two $m \times n$ matrices with entries in $F$ . Recall that in the category ${Matr}_{F}$ these matrices correspond to two arrows $n \to m$ .

Describe the equalizer of $A, B : n \to m$ in ${Matr}_{F}$ .
Describe the coequalizer of $A, B : n \to m$ in ${Matr}_{F}$ .

Hints

First show that the equalizer should be a matrix $E$ with $n$ rows such that $A E = B E$ , such that for every other matrix $C$ with $n$ rows and $A C = B C$ there exists a unique factorization $C = E H$ . Then investigate what you can say about the columns of a matrix $C$ that satisfies $A C = B C$ . Show that this condition is equivalent to the condition that every column of $C$ is contained in the null space of $A - B$ . Now suppose ${e_{1}, \dots, e_{k}}$ is a basis for the null space of $A - B$ . To say $c_{i}$ is contained in the null space of $A - B$ then means there is a unique $F$ -linear combination with $c_{i} = h_{1, i} e_{1} + \dots h_{k, i} e_{k}$ . The matrix $C$ should reveal to be related to the matrix $E$ By way a matrix $H$ ...