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# Matrices

## Definition of a matrix

A matrix is a table or array of numbers or letters arranged in rows and columns. The table is delimited by brackets [ ] or parentheses ( ). The plural of matrix is matrices.

These are four examples of matrices of different sizes: $$\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad \begin{bmatrix}1&x\\1&y\\1&z \end{bmatrix},\quad \begin{pmatrix}1&1\\0&1\end{pmatrix},\quad \begin{bmatrix}1\\0\\-1\end{bmatrix}$$

The numbers contained within a matrix are called elements or entries.

## Order of a matrix

The order of a matrix is written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns. This is read as "$m$ by $n$". The order is also called the size of the matrix.

The matrix $\displaystyle\begin{pmatrix}0&1&0\\-1&0&0\end{pmatrix}$ is a $2\times 3$ matrix. It has $2$ rows, $3$ columns and $6$ entries.

We can find the number of elements by multiplying $m$ and $n$. Even though an $m \times n$ and an $n \times m$ matrix have the same number of elements, they do not have the same order.

A $2\times 3$ matrix and a $3\times 2$ matrix have different sizes though they have same number of elements (6).

The following matrices $$\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad \begin{bmatrix}1&x\\1&y\\1&z \end{bmatrix},\quad \begin{pmatrix}1&1\\0&1\end{pmatrix},\quad \begin{bmatrix}1\\0\\-1\end{bmatrix}$$ are of order $2\times 3$, $3\times2$, $2\times 2$ and $3\times 1$.

## Square, row and column matrices

There are special type of matrices depending on their size.

• A square matrix has the same number of rows and columns.
• A row matrix is a matrix with one row only. It is also called a row vector.
• A column matrix is a matrix with one column only. It is also called a column vector.

Examples of a square matrix, a row vector and a column vector:

$$\begin{pmatrix}1&2\\-1&0\end{pmatrix},\quad \begin{pmatrix}1&2\end{pmatrix},\quad \begin{pmatrix}1\\0\\3\end{pmatrix}.$$
A square matrix

## Zero and identity matrices

There are distinctive types of matrices depending on their elements.

• A zero matrix is a matrix with zeroes everywhere.
• An identity matrix is a square matrix with ones on the main diagonal (upper left to lower right) and zeroes off the diagonal. This matrix is usually given the symbol $I$.

Examples of a zero matrix and an identity matrix.

$$\begin{pmatrix}0&0\\0&0\\0&0 \end{pmatrix},\quad \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}.$$

## Multiplication of a matrix by a number

To multiply a matrix by a number we multiply all the entries of the matrix by this number. This number is often called a scalar.

$$\Tred{2}\begin{pmatrix}1&2\\-1&0\end{pmatrix} = \begin{pmatrix}\Tred{2}\times 1&\Tred{2}\times 2\\\Tred{2}\times -1&\Tred{2}\times 0\end{pmatrix} = \begin{pmatrix}2&4\\-2&0\end{pmatrix}$$

To add two matrices, we add each entry of one matrix to the corresponding entry of the matrix.

$$\begin{pmatrix}\Tred{1}&\Tblue{2}\\\Tgreen{-1}&\Torange{0}\end{pmatrix} + \begin{pmatrix}\Tred{-3}&\Tblue{1}\\\Tgreen{1}&\Torange{2}\end{pmatrix} = \begin{pmatrix}\Tred{1-3}&\Tblue{2+1}\\\Tgreen{-1+1}&\Torange{0+2}\end{pmatrix} = \begin{pmatrix}\Tred{-2}&\Tblue{3}\\\Tgreen{0}&\Torange{2}\end{pmatrix}$$

The two matrices must have the same size. The resulting matrix has the same size.

The sum below is NOT defined because the two matrices don't have the same size. $$\begin{pmatrix}1&2\\-1&3\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix}$$

## Multiplication of two matrices

The multiplication $A\times B$ of two matrices $A$ and $B$ is not straightforward. We assume that $A$ has size $m_A\times n_A$ and $B$ has size $m_B\times n_B$.

• Size requirement. We must have $$n_A = m_B$$
• Size of the product The matrix $A\times B$ has size $m_A\times n_B$..
• To compute one entry, we isolate one row in $A$ and one column in $B$. We multiply each pairing terms and we add them. We repeat the process for each row of $A$ and each column of $B$.

The following products are not defined because the matrices have incompatible sizes.

$$\begin{pmatrix} 1&0\\ 1&2\\ 0&1 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 0&1\\ 1&0 \end{pmatrix},\; \begin{pmatrix} 1&0 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 0&1\\ 1&0 \end{pmatrix},\; \begin{pmatrix} 1\\ 0 \end{pmatrix} \times \begin{pmatrix} 1\\ 0 \end{pmatrix}.$$

## Order matters for matrix multiplication

Unlike the multiplication we do with real numbers, the order in which we multiply matrices is important. We say that the matrix product is noncommutative.

• Size: If $A$ has size $2\times3$ and $B$ has size $3\times1$, we can compute $A\times B$, because the number of columns of $A$ is equal to the number of rows of $B$. But we cannot compute $B\times A$, because the number of columns of $B$ is not equal to the number of rows of $A$.
• Order: When the matrices are square and have the same size, we can compute $A\times B$ and $B\times A$. However the resulting matrices will usually not be the same.
\begin{align*} A\times B = &\begin{pmatrix}1&1\\0&1\end{pmatrix}\times \begin{pmatrix}1&0\\1&1\end{pmatrix}= \begin{pmatrix}2&1\\1&1\end{pmatrix},\\ B\times A = &\begin{pmatrix}1&0\\1&1\end{pmatrix}\times \begin{pmatrix}1&1\\0&1\end{pmatrix}= \begin{pmatrix}1&1\\1&2\end{pmatrix} \end{align*}

## Summary of matrix operations

We summarise the matrix operations for the number $x = 2$ and the square matrices $$A = \begin{pmatrix} 0&1\\ 1&2 \end{pmatrix},\quad\; B = \begin{pmatrix} 1&2\\ 0&-1 \end{pmatrix}$$
Operation Notation Computation Scalar multiplication Addition Multiplication $$x A$$ $$A+B$$ $$A\times B$$ $$\begin{pmatrix} 0&2\\ 2&4 \end{pmatrix}$$ $$\begin{pmatrix} 1&3\\ 1&1 \end{pmatrix}$$ $$\begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}$$ None Matrices must be the same size Number of columns of $A$ = number of rows of $B$