Equal Matrices #
Definition: #
Let \(\bf A\) and \(\bf B\) be two matrices of same order such that \(\bf A=[A_{ij}]_{m \times n}\) and \(\bf B=[B_{ij}]_{m \times n}\). Then the matrices \(\bf A\) and \(\bf B\) are said to be \(\fcolorbox{red}{white}{\bf equal matrices}\) iff $$ \bf A_{ij}=B_{ij} $$
Multiplication by a Scalar to a Matrix #
Definition: #
Let \(\bf A=[A_{ij}]_{m \times n}\) be an \(\bf m \times n\) matrix and \(\bf k\) be scalar then the \(\fcolorbox{red}{white}{\bf multiplication of the scalar}\) \(\bf k\) to the matrix \(\bf A\) is denoted by \(\fcolorbox{red}{white}{\bf \(\bf kA\)}\) and is defined by the matrix \(\bf C\) of order \(\bf m \times n\) such that $$\bf kA=C=[C_{ij}]_{m \times n} $$ where $$\bf C_{ij}=kA_{ij} $$
Addition of Two Matrices #
Definition: #
Let \(\bf A\) and \(\bf B\) be two matrices of same order such that \(\bf A=[A_{ij}]_{m \times n}\) and \(\bf B=[B_{ij}]_{m \times n}\). Then the \(\fcolorbox{red}{white}{\bf addition the matrices}\) \(\bf A\) and \(\bf B\) is denoted by \(\fcolorbox{red}{white}{\bf \(\bf A+B\)}\) and is defined by the matrix \(\bf C\) of order \(\bf m \times n\) such that $$\bf A+B=C=[C_{ij}]_{m \times n} $$ where $$\bf C_{ij}=A_{ij}+B_{ij} $$
Multiplication of Two Matrices #
Definition: #
Let \(\bf A=[A_{ij}]_{m \times n}\) be an \(\bf m \times n\) matrix and \(\bf B=[B_{ij}]_{n \times p}\) be an \(\bf n \times p\) matrix. Then the \(\fcolorbox{red}{white}{\bf product}\) of \(\bf A\) and \(\bf B\), is denoted by \(\fcolorbox{red}{white}{\bf \(\bf AB\)}\), and is defined by the matrix \(\bf C\) of order \(\bf m \times p\) such that $$\bf AB=C=[C_{ij}]_{m \times p}$$ where $$\bf C_{ij}=\displaystyle\sum_{p=1}^{n}A_{ip}B_{pj}$$
