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Matrices

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.

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 $$3\times 1$$.

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
A square matrix

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.

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

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. $$$ \begin{pmatrix}1&2\\-1&3\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix} $$$

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 $$B$$.
\begin{align*} & \times \begin{pmatrix} \qquad\quad\Tred{1}\qquad&\qquad\quad\Torange{1}\qquad\quad\\ \qquad\quad\Tred{2}\qquad&\qquad\quad\Torange{4}\qquad\quad \end{pmatrix}\\ \begin{pmatrix} \Tblue{1}&\Tblue{2}\\ \Tgreen{1}&\Tgreen{3}\\ \Tviolet{1}&\Tviolet{0} \end{pmatrix} &= \begin{pmatrix} \Tblue{1}\times\Tred{1} + \Tblue{2}\times \Tred{2}& \Tblue{1}\times\Torange{1} + \Tblue{2}\times\Torange{4}\\ \Tgreen{1}\times\Tred{1} + \Tgreen{3}\times \Tred{2}& \Tgreen{1}\times\Torange{1} + \Tgreen{3}\times\Torange{4}\\ \Tviolet{1}\times\Tred{1} + \Tviolet{0}\times \Tred{2}& \Tviolet{1}\times\Torange{1} + \Tviolet{0}\times\Torange{4} \end{pmatrix}\\ &= \begin{pmatrix} \qquad\quad5\qquad&\qquad\quad9\qquad\quad\\ \qquad\quad7\qquad&\qquad\quad13\qquad\quad\\ \qquad\quad1\qquad&\qquad\quad1\qquad\quad \end{pmatrix} \end{align*}

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

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

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*}

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 Scalar multiplication Addition Multiplication
Notation $$$x A$$$ $$$A+B$$$ $$$A\times B$$$
Computation $$$\begin{pmatrix} 0&2\\ 2&4 \end{pmatrix}$$$ $$$\begin{pmatrix} 1&3\\ 1&1 \end{pmatrix}$$$ $$$\begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}$$$
Restrictions None Matrices must be the same size Number of columns of $$A$$ = number of rows of $$B$$