2.2 Types of Matrices

Here we will consider some special types of matrices.

1.

The zero matrix is a matrix in which all the entries are zeroes. An example of zero matrix is \( \begin {pmatrix} 0&0&0\\0&0&0\\ \end {pmatrix} \)

2.

A matrix in which the number of rows equals the number of columns is called a square matrix \( \begin {pmatrix} 1&2\\3&4\\ \end {pmatrix} \) and \( \begin {pmatrix} 1&2&1\\4&2&-1\\3&-3&7\\ \end {pmatrix} \) are squares matrices.

3.

A square matrix is called a diagonal matrix if all its non-diagonal entries are zeroes. The matrix \( \begin {pmatrix} 1&0&0\\0&2&0\\0&0&3\\ \end {pmatrix} \) is a diagonal matrix.

4.

An identity matrix is a square matrix whose diagonal entries are all ones and the non-diagonal entries are zeroes \( \begin {pmatrix} 1&0\\ 0&1 \end {pmatrix} \) and \( \begin {pmatrix} 1&0&0\\ 0&1&0\\0&0&1\\ \end {pmatrix} \) are identity matrices.

5.

A square matrix in which all the entries below that leading diagonal are zeroes is called an upper triangular matrix. A lower triangular matrix is a square matrix in which all the elements above the leading diagonal are zeroes \( \begin {pmatrix} 6&3&1\\0&2&7\\0&0&1\\ \end {pmatrix} \) is upper triangular and \( \begin {pmatrix} 4&0&0\\4&2&0\\7&1&-3\\ \end {pmatrix} \) is lower triangular.

6.

The transpose of a matrix \(A\), denoted by \(A^t\) is the matrix obtained by interchanging the rows and columns of \(A\) if \(A= \begin {pmatrix} 1&3&4\\-1&0&2\\ \end {pmatrix} \), the \( A^t= \begin {pmatrix} 1&-1\\3&0\\4&2\\ \end {pmatrix} \)

7.

A matrix \(A\) is said to be symmetric if \(A=A^t\), that is, if \(a_{ij}=a_{ji}\) — the entries are mirrored in the leading diagonal.

8.

A matrix \(A\) is said to be skew-symmetric if \(A=-A^t\), that is, if \(a_{ij}=-a_{ji}\). The diagonal of such a matrix is necessarily zero: taking \(i=j\) gives \(a_{ii}=-a_{ii}\), so \(a_{ii}=0\).

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