7.01 Matrix review

In mathematics, a matrix is a particular method of displaying information. It is any rectangular array of numbers, symbols, or expressions arranged in rows and columns. So the table above would be represented by a matrix, which we can call A and is shown below.A=\begin{bmatrix} 13 & 6 \\ 3 & 10 \\ 8 & 9 \end{bmatrix}

We refer to the dimensions or order of a matrix as a reference to the number of rows and number of columns.

A matrix with dimensions m\times n has m rows and n columns. For instance, the following matrix has dimensions 3\times 4:\begin{bmatrix} 3 & 4 &5&3 \\ 1 & 0 & 7&0 \\ 6 & 0 &4 &0 \end{bmatrix}

Elements are the individual entries of a matrix. An element can be identified by its position (that is, its row and column) in the matrix. For the following matrix B, the elements in the second row and third column is 7, where we use the following notation b_{23}=7.B=\begin{bmatrix} 1&3&0&1 \\ 4&5&7&8 \end{bmatrix}

Generally, we may represent any matrix with m rows and n columns as shown:A=\begin{bmatrix} a_{11}&a_{12}&a_{13}&a_{14}&...&a_{1n} \\ a_{21}&a_{22}&a_{23}&a_{24}&...&a_{2n} \\ .&.&.&.&...&. \\ .&.&.&.&...&. \\ .&.&.&.&...&. \\ a_{m1}&a_{m2}&a_{m3}&a_{m4}&...&a_{mn} \end{bmatrix}

A row matrix or row vector has just a single row. The following matrix T is an example of a row matrix.T=\begin{bmatrix} 3&4&5 \end{bmatrix}

A column matrix or column vector has just a single column. The following matrix M is an example of a column matrix.M=\begin{bmatrix} 5\\3\\0\\11 \end{bmatrix}

A square matrix has an equal number of rows and columns. The matrices G and J are examples of square matrices.G=\begin{bmatrix} 1&5\\ 5&19 \end{bmatrix}, \quad J=\begin{bmatrix} 2&0&-1\\ 16&\sqrt{2}&5\\ 0.1&3&-7 \end{bmatrix}

A diagonal matrix is a square matrix with zero elements off of the leading diagonal or main diagonal. The leading diagonal represents the elements along the diagonal starting at the top-left to the bottom-right. The matrix C is an example of a diagonal matrix.C=\begin{bmatrix} 6&0&0\\ 0&-7&0\\ 0&0&\frac{4}{7} \end{bmatrix}

An identity matrix is a special type of diagonal matrix where all the elements on the main diagonal are ones.

For example: N = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}

Matrix N is also called a binary matrix as it consists entirely of 1's and 0's.

A zero matrix is a matrix of any dimension where all of the elements are zero. For example: \begin{pmatrix} 0&0\\ 0&0 \end{pmatrix}

A triangular matrix is a square matrix where:

• All entries above the diagonal are zero (called a lower triangular matrix)

• OR all entries below the diagonal are zero (called an upper triangular matrix)

• OR the entries above and below the diagonal are zero (called a diagonal matrix)

• OR all of the entries are zero (called a zero matrix)

Examples

Two matrices are said to be equal if every corresponding elements in the matrices are equal.

\begin{bmatrix} 3 & 4 \\ -2 & 7 \end{bmatrix} = \begin{bmatrix} m & 4 \\ -2 & n \end{bmatrix}

So in this case, since the two matrices are equal, then m=3 and n=7.

Examples

Simultaneous equations follow the convention of organising the coefficients into matrix notation. For instance, consider the two equations x+2y=7 and 2x-5y=-4. This system can be written like this using matrices.

\begin{bmatrix} 1 & 2 \\ 2 & -5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -4 \end{bmatrix}

Here is a network showing roads between towns. The information can be put into a matrix to make it easier to read.

We can construct a matrix, where the entries represent the number of paths between the towns.

We will need to set up a 5\times 5 matrix, where each of the rows and columns represents a town:\begin{array}{cc} & \begin{array}{ccc} 1 & 2 & 3 & 4& 5 \end{array} \\ \begin{array}{c} 1 \\ 2 \\ 3\\4\\5 \end{array} & \left[ \begin{array}{ccc} . & . & . &.&. \\ . & . & . &.&. \\ . & . & . &.&.\\ . & . & . &.&.\\ . & . & . &.&. \end{array}\right] \end{array}

The next step is to fill in the numbers of roads between them. There are two roads between towns 1 and 3, so we input a 2 in a_{13} and a_{31}. As there are no roads connecting town 2 with any other towns, all elements in row 2 and column 2 are 0. \begin{bmatrix} 0&0&2&1&0\\ 0&0&0&0&0\\ 2&0&0&0&1\\ 1&0&0&0&3\\ 0&0&1&3&0 \end{bmatrix}

Since the rows and columns in the two way frequency table correspond to the rows and columns of the matrix, we can simply transfer the information into a matrix: \begin{bmatrix} 16&22\\ 10&19\\ 26&41 \end{bmatrix}

Examples