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 ${\textbf {A}}$ and ${\textbf {B}}$ both be $2\times 2$ 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 ${\textbf {A}}$ and ${\textbf {B}}$ be $n\times m$ matrices, we would have:

${\begin{aligned}{\textbf {A}}+{\textbf {B}}&={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{pmatrix}}\end{aligned}}\,.$

Example

${\begin{pmatrix}1&3\\2&1\\\end{pmatrix}}+{\begin{pmatrix}4&0\\1&5\\\end{pmatrix}}={\begin{pmatrix}5&3\\2&1\\\end{pmatrix}}$

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 $\mathbb {R} ^{n}$, so our scalars will be elements from $\mathbb {R}$.

Scalar multiplication acts by multiplying each component of the matrix by the same scalar. Letting $\lambda$ be an element of our field and ${\textbf {A}}$ be a $2\times 2$ matrix, we have:

$\lambda \mathbf {A} =\lambda {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{pmatrix}}={\begin{pmatrix}\lambda a_{11}&\lambda a_{12}\\\lambda a_{21}&\lambda a_{22}\\\end{pmatrix}}\,.$

If we instead let ${\textbf {A}}$ be a more general $n\times m$ matrix, we would have:

$\lambda \mathbf {A} =\lambda {\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}={\begin{pmatrix}\lambda A_{11}&\lambda A_{12}&\cdots &\lambda A_{1m}\\\lambda A_{21}&\lambda A_{22}&\cdots &\lambda A_{2m}\\\vdots &\vdots &\ddots &\vdots \\\lambda A_{n1}&\lambda A_{n2}&\cdots &\lambda A_{nm}\\\end{pmatrix}}\,.$

Examples

$3{\begin{pmatrix}2&1\\3&5\\\end{pmatrix}}={\begin{pmatrix}6&3\\9&15\\\end{pmatrix}}$