Fandom

Math Wiki

Matrix

1,010pages on
this wiki
Add New Page
Talk0 Share

A matrix is a table (or an array) of objects, such as quantities, arranged in rows and columns.

One can represent an entry in a matrix as a_{i,j} where the subindex i walks the rows and j walks the columns. For example a_{3,4} would be the entry in row 3 and column 4.

Here is an example of a matrix (a 3\times4 matrix with integer entries):

\begin{bmatrix}7&341&110&14\\-27&34&0&-19\\20&26&33&25\end{bmatrix}

Matrices can be formed with other kinds of objects.

Here is an example using polynomials for the entries.

\begin{bmatrix}x^2-1&11x+20\\x^3&x^4+4x^3+8\end{bmatrix}

This is a 2\times2-matrix using polynomials of one variable.

Matrices can also be used to model translations from \R^n\to\R^m .

Formal defintion

Let m,n\in\N .
Let S be a set.
Then a m\times n matrix over S is a mapping a:[1,m]\times[1,n]\to S .

Matrix operations

Matrix addition and subtraction

Matrix addition and subtraction can only be done on matrices of the same dimensions. Corresponding entries are added (or subtracted).


\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\end{bmatrix}+
\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\end{bmatrix}=
\begin{bmatrix}a_{1,1}+b_{1,1}&a_{1,2}+b_{1,2}\\a_{2,1}+b_{2,1}&a_{2,2}+b_{2,2}\end{bmatrix}

Ordinary matrix multiplication

The ordinary matrix product is the most often used and the most important way to multiply matrices. It is defined between two matrices only if the width of the first matrix equals the height of the second matrix. Multiplying an m\times n matrix with an n\times p matrix results in an m\times p matrix. The ordinary matrix product is not commutative.


\begin{bmatrix}a_{1,1}&a_{1,2}&a_{1,3}\\a_{2,1}&a_{2,2}&a_{2,3}\end{bmatrix}
\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\b_{3,1}&b_{3,2}\end{bmatrix}
=
\begin{bmatrix}
a_{1,1}\cdot b_{1,1}+a_{1,2}\cdot b_{2,1}+a_{1,3}\cdot b_{3,1}&a_{1,1}\cdot b_{1,2}+a_{1,2}\cdot b_{2,2}+a_{1,3}\cdot b_{3,2}\\
a_{2,1}\cdot b_{1,1}+a_{2,2}\cdot b_{2,1}+a_{2,3}\cdot b_{3,1}&a_{2,1}\cdot b_{1,2}+a_{2,2}\cdot b_{2,2}+a_{2,3}\cdot b_{3,2}
\end{bmatrix}

Scalar matrix multiplication

The scalar matrix product is the result of multiplying a matrix by a scalar. This entails nothing more than multiplying each entry in the matrix by the value of the scalar. For example, letting r be a scalar and A be a matrix,

\mathbf{A}={\begin{bmatrix}a&b\\c&d\end{bmatrix}}

r\mathbf{A}=r{\begin{bmatrix}a&b\\c&d\end{bmatrix}=
\begin{bmatrix}r\cdot a&r\cdot b\\r\cdot c&r\cdot d\end{bmatrix}}

Misc

To remove the last row of an m\times n matrix, multiply by the m\times m identity matrix augmented with a m\times n 0 matrix.

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.