What are matrix coefficients in linear algebra? What are matrix coefficients in linear algebra? And what does it mean "integer matrix coefficients"?
 A: Note: there is a big difference between the terms "matrix coefficient" and "coefficient matrix". I'll explain first what you are probably asking about:
Coefficient matrix
Suppose you have a system of equations:
$$\begin{align*}
1\cdot x_1 + 2x_2 &= 16\\
3x_1 + 1\cdot x_2 &= 4 \\
\end{align*}
\tag{1}$$
$(I)$ Then the coefficient matrix (in this case, with integer entries) corresponding to this system of linear equations in $(1)$ is:
$$M = 
\begin{bmatrix}
1 & 2\\
3 & 1\\
\end{bmatrix}
$$
where the entries in first column represents the coefficients of the $x_1$, and those in the second column the coefficients of $x_2$, etc..
The augmented coefficient matrix $M_a$ would include the entries in a third column which correspond to the values at the right of the equals signs in $(1)$: 
$$M_a = 
\begin{bmatrix}
1 & 2 &\;|\; 16\\
3 & 1 &|\; 4\;\\
\end{bmatrix}
$$

Matrix coefficient
$(2)$ On the other hand, this coefficient matrix contrasts with what is meant by a matrix coefficient. (Please read more at the given linked entry from Wikipedia: what follows is a brief excerpt from that entry.)

In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding matrices.

A: Matrix coefficients are the entries of the matrix. Integer matrix coefficients is just a matrix with integer entries.
