Matrix Operations

Matrix Addition

Matrix addition is component-wise. Thus, in order to add two matrices together, they must have the same number of rows and the same number of columns. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\textbf {A}}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\textbf {B}}} both be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\times 2} matrices. Then:

${\textbf {A}}+{\text{B}}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\\\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\\\end{pmatrix}}\,.$

If instead we let both Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\textbf {A}}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\textbf {B}}} be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n\times m} matrices, we would have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}{\textbf {A}}+{\textbf {B}}&={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{pmatrix}}\end{aligned}}\,.}

Example

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{pmatrix}1&3\\2&1\\\end{pmatrix}}+{\begin{pmatrix}4&0\\1&5\\\end{pmatrix}}={\begin{pmatrix}5&3\\2&1\\\end{pmatrix}}}

Scalar Multiplication

We can also scale a matrix by multiplying it by a scalar. Since a matrix's rows or columns are vectors in a vector field, the scalar will be an element of the underlying field, i.e. an element of the same type as each component of the matrix. We will often assume that our vector field is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{n}} , so our scalars will be elements from $\mathbb {R}$.

Scalar multiplication acts by multiplying each component of the matrix by the same scalar. Letting Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda } be an element of our field and ${\textbf {A}}$ be a Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\times 2} matrix, we have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda \mathbf {A} =\lambda {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{pmatrix}}={\begin{pmatrix}\lambda a_{11}&\lambda a_{12}\\\lambda a_{21}&\lambda a_{22}\\\end{pmatrix}}\,.}

If we instead let ${\textbf {A}}$ be a more general Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n\times m} matrix, we would have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda \mathbf {A} =\lambda {\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}={\begin{pmatrix}\lambda A_{11}&\lambda A_{12}&\cdots &\lambda A_{1m}\\\lambda A_{21}&\lambda A_{22}&\cdots &\lambda A_{2m}\\\vdots &\vdots &\ddots &\vdots \\\lambda A_{n1}&\lambda A_{n2}&\cdots &\lambda A_{nm}\\\end{pmatrix}}\,.}

Examples

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3{\begin{pmatrix}2&1\\3&5\\\end{pmatrix}}={\begin{pmatrix}6&3\\9&15\\\end{pmatrix}}}