Linear Algebra – Matrices

\[A=\Bigg(\mathbf{a}_1 \ldots \mathbf{a}_n\Bigg)=\begin{pmatrix}a_{11}&\ldots&a_{1n}\\\vdots&&\vdots\\a_{m1}&\ldots&a_{mn}\end{pmatrix} \quad\text{is an}\quad m\times n \text{matrix}.\]

Here is \(m\) the number of rows and \(n\) the number of columns of \(A\). Moreover, \(m\times n\) is the size of the matrix \(A\). If the size is clear, then we shortly write \(A=(a_{ij})\), where \(a_{ij}\) denotes the entry of \(A\) in the \(i^{th}\) row and in the \(j^{th}\) column.

The entries \(a_{11},a_{22},a_{33},\ldots\) are called the diagonal entries of \(A\); they form the so-called main diagonal of \(A\).

A diagonal matrix is a square matrix (so: \(m=n\)) for which all nondiagonal entries are zero.

A matrix for which all entries are zero, is called a zero matrix. Notation: \(0\).

Theorem: Let \(A\), \(B\), \(C\) and \(0\) be matrices of the same size, and let \(r\) and \(s\) be scalars, then we have:

\(A+B=B+A\) \((A+B)+C=A+(B+C)\) \(A+0=A\)

\(r(A+B)=rA+rB\) \((r+s)A=rA+sA\) \(r(sA)=(rs)A\)

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Last modified on March 1, 2021