# What are matrix coefficients in linear algebra?

What are matrix coefficients in linear algebra? And what does it mean "integer matrix coefficients"?

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If you read this term from some sentence in some literature, quoting the sentence in your question would be helpful for the readers to understand the context. – user1551 Dec 29 '12 at 19:11

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

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Thanks! Didn't know the definition. – J-unior Dec 29 '12 at 18:24
@J-unior Glad to help. Remember you always can upvote posts which are useful to you. You can accept the answer is you consider it really answers your question. – leo Dec 29 '12 at 18:29
I know, I'm not new at stackexchange. Still don't have enough rep to do this, and 10 minutes are not over yet :-) – J-unior Dec 29 '12 at 18:31

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.

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