Basics of matrices

Matrices are rectangular arrays of objects, defined using specific terminology such as rows, columns, order, and elements. Understanding this terminology is essential for performing basic operations like addition, subtraction, scalar multiplication, and matrix multiplication.

Use this page to revise the following concepts of matrices:

Matrix Definitions

A matrix denoted by an uppercase pronumeral such as \(A\), can be represented as:

\[A=\left[\begin{align}& a_{11}\space\space a_{12}\space\space a_{13}\\& a_{21}\space\space a_{22}\space\space a_{23}\end{align}\right]\]

The elements are the entries within the matrix, are represented by a corresponding lowercase pronumeral, such as \(a_{ij}\) where \(i\) refers to the row, and \(j\) refers to the column.

The order of a matrix refers to its size, defined by the number of rows and columns the matrix contains. This is written as \(r\times c\) and pronounced as "r by c."

For example, the matrix above is a \(2\times3\) matrix, meaning that it has 2 rows and 3 columns.

Special Types of Matrices

There are special types of matrices with unique properties that are important for understanding how matrices can be applied in specific contexts, such as identity matrices in solving systems of linear equations and diagonal matrices in simplifying computations.

There are also two fundamental matrix forms:

Basic Matrix Operations

Addition and Subtraction of Matrices

Matrix addition and subtraction are performed element by element. This means each element in one matrix is added to or subtracted from the corresponding element in the other matrix.

Note

For these operations to be valid, both matrices must have the same order.

For the following matrices \(A\) and \(B\):

\(A=\left[\begin{align}& a_{11}\space\space a_{12}\\& a_{21}\space\space a_{22}\end{align}\right]\)               \(B=\left[\begin{align}& b_{11}\space\space b_{12}\\& b_{21}\space\space b_{22}\end{align}\right]\)

\(A+B\) and \(A-B\) are both valid as \(A\) and \(B\) are both \(2\times 2\) matrices.

\[A+B=\left[\begin{align}& a_{11} + b_{11}\space\space a_{12}+b_{12}\\& a_{21} + b_{21}\space\space a_{22} + b_{22}\end{align}\right]\]

\[A-B=\left[\begin{align}& a_{11} - b_{11}\space\space a_{12} - b_{12}\\& a_{21} - b_{21}\space\space a_{22} - b_{22}\end{align}\right]\]


Worked Example

For the following matrices:

\(A=\left[\begin{align}& 1\space\space 2\space\space 3\\& 4\space\space 5\space\space 6\end{align}\right]\)        \(B=\left[\begin{align}& 1\space\space 2\\& 3\space\space 4\end{align}\right]\)        \(C=\left[\begin{align}& 1\space\space 3\space\space 5\\& 2\space\space 4\space\space 6\end{align}\right]\)

Determine which of the following matrix additions is valid. If valid, calculate the sum.

\(A+B\) is not valid, as \(A\) is a \(2\times 3\) matrix, whereas \(B\) is a \(2\times 2\) matrix. However, both additions and subtraction between \(A\) and \(C\) are valid, as both \(A\) and \(C\) are \(2\times 3\) matrices.

Therefore \(A+C\) is valid:

\(A+C=\left[\begin{align}& 1\space\space 2\space\space 3\\& 4\space\space 5\space\space 6\end{align}\right]\) \(+\) \( \left[\begin{align}& 1\space\space 3\space\space 5\\& 2\space\space 4\space\space 6\end{align}\right]\)\(=\)\(\left[\begin{align}& 1+1\space\space 2+3\space\space 3+5\\& 4+2\space\space 5+4\space\space 6+6\end{align}\right]\)\(=\) \(\left[\begin{align}& 2\space\space 5\space\space 8\\& 6\space\space 9\space\space 12\end{align}\right]\)

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a constant (number). This operation scales every element while maintaining the matrix's original order.

For matrix \(A\):

\[A=\left[\begin{align}& a_{11}\space\space a_{12}\\& a_{21}\space\space a_{22}\end{align}\right]\]

Multiplying it by a scalar \(k\)

\[kA=\left[\begin{align}& ka_{11}\space\space ka_{12}\\& ka_{21}\space\space ka_{22}\end{align}\right]\]



Worked Example

For the following matrix:

\(A=\left[\begin{align}& 1\space\space 2\space\space 3\\& 4\space\space 5\space\space 6\end{align}\right]\)

Find \(3A\).

\(3A=\left[\begin{align}& 3\times1\space\space 3\times2\space\space 3\times3\\& 3\times4\space\space 3\times5\space\space 3\times6\end{align}\right]\) \(=\) \( \left[\begin{align}& 3\space\space\space\space\space 6\space\space\space\space\space 9\\& 12\space\space 15\space\space 18\end{align}\right]\)

Transpose Matrix

The Transpose of a matrix \(A\) is denoted by \(A^{T}\), formed by swapping the rows and columns of \(A\).

Elements \(a_{ij}\) in \(A\) become \(a_{ji}\) in \(A^{T}\)

Likewise, the order of the matrix will also swap. If \(A\) is an \(n\times m\) matrix, then \(A^{T}\) is an \(m\times n\) matrix.

The matrices below show a matrix \(A\) and its transpose \(A^{T}\):

\[A=\left[\begin{align}& 1\space\space 2\space\space 3\\& 4\space\space 5\space\space 6\end{align}\right]\]

\[A^{T}=\left[\begin{align}& 1\space\space 4\\& 2\space\space 5\\& 3\space\space 6\end{align}\right]\]