What are matrix coefficients in linear algebra? And what does it mean "integer matrix coefficients"?
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$\begingroup$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.)
$\endgroup$ $\begingroup$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.
Matrix coefficients are the entries of the matrix. Integer matrix coefficients is just a matrix with integer entries.
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