Row matrix

A matrix having only one row and any number of columns is called a row matrix or a row vector.

Column Matrix

A matrix having only one column and any number of rows is called a column matrix or a column vector.

Zero Matrix or Null Matrix

A matrix, each of whose elements is zero, is called a zero matrix or a null matrix.

For example, the matrices \(\left[ {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right],\,\,\,\left[ {\begin{array}{*{20}{c}}0&0&0\\0&0&0\end{array}} \right]\) are null matrices of order 2×2 and 2×3 respectively.

Square matrix

A matrix in which the number of rows is equal to the number of columns, is called a square matrix.

Diagonal matrix

A square matrix \(A = {\left[ {{a_{ij}}} \right]_{m \times n}}\)is called a diagonal matrix if all the elements except those in the leading diagonal are zero, i.e. \({a_{ij}} = 0\) for \(i \ne j\). In other words\(A = {\rm{diag}}.\left[ {{a_{11}}{a_{22}}{a_{33}} \ldots {a_{mn}}} \right]\)

Note that the main diagonal is a set of all elements given by \({a_{ii}}\) . You can see it's possible in a rectangular matrix also. See the below example.

\(\left[ {\begin{array}{*{20}{c}}2&0&0\\0&3&0\end{array}} \right]\)

This rectangular matrix satisfies this condition as \({a_{11}} = 2,\,\,{a_{22}} = 3\) and rest of the elements are zero. Therefore, a diagonal matrix may not always be a square matrix. It can be a rectangular matrix as well.

Scalar Matrix

A square matrix in which every non-diagonal element is zero and all diagonal elements are equal, is known as a scalar matrix.

For example, \(\left[ {\begin{array}{*{20}{c}}\lambda &0&0\\0&\lambda &0\\0&0&\lambda \end{array}} \right]\) is a scalar matrix.

Unit Matrix

a square matrix in which every non-diagonal element is zero and every diagonal element is 1 is called a unit matrix or an identity matrix. Thus, a square matrix \(A = {\left[ {{a_{ij}}} \right]_{m \times n}}\)is a unit matrix if

\({a_{ij}}\,\, = \,\,\left\{ {\begin{array}{*{20}{c}}{0\,\,\,\,{\rm{when}}\,\,\,i\, \ne \,j}\\{1\,\,\,\,{\rm{when}}\,\,\,i = j}\end{array}} \right.\)

A unit matrix of order \(n\)is denoted by \({I_n}\) or\(I\). For example, \({I_2} = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\) and \({I_3} = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\) are unit matrices of order 2 and 3 respectively.

Upper Triangular Matrix

A square matrix \(A = \left[ {{a_{ij}}} \right]\)is called upper triangular matrix if \({a_{ij}} = 0\) for all \(i > j\). for example, the matrix  A = \(\left[ {\begin{array}{*{20}{c}}1&2&3\\0&3&2\\0&0&1\end{array}} \right]\)is an upper triangular matrix.

Lower Triangular matrix

A square matrix \(A = \left[ {{a_{ij}}} \right]\) is called a lower triangular matrix if \({a_{ij}} = 0\) for all \(i < j\). For example the matrix  \(A = \left[ {\begin{array}{*{20}{c}}2&0&0&0\\3&4&0&0\\1&3&5&0\\2&4&6&7\end{array}} \right]\) is a lower triangular matrix.

Trace of a Matrix

The sum of the diagonal elements of a square matrix A is called the trace of A and is denoted by trace (A). For example, if

\(A = \left[ {\begin{array}{*{20}{c}}1&3&2\\2&{ - 1}&0\\3&1&5\end{array}} \right],\) then trace (A) = 1 – 1 + 5 = 5.

Sub Matrix

A matrix that is obtained from a given matrix by deleting any number of rows or columns or both is called a sub-matrix of the given matrix. For example,  \(\left[ {\begin{array}{*{20}{c}}2&{ - 1}\\3&5\end{array}} \right]\) is a submatrix of \(\left[ {\begin{array}{*{20}{c}}5&3&1\\4&2&{ - 1}\\6&3&5\end{array}} \right].\)

EQUALITY OF MATRICES 

Two matrices A and B are said to be equal if they are of the same order and all the corresponding elements are equal. It is written as A = B. For example,

\(A = \left[ {\begin{array}{*{20}{c}}2&3&4\\5&1&0\end{array}} \right]\) and  \(B = \left[ {\begin{array}{*{20}{c}}2&{1 + 2}&{1 + 3}\\{2 + 3}&1&0\end{array}} \right]\) are equal matrices, whereas

\(C = \left[ {\begin{array}{*{20}{c}}3&4\\2&1\end{array}} \right]\) and  \(D = \left[ {\begin{array}{*{20}{c}}2&5&1\\2&3&1\end{array}} \right]\)  are not equal, because their orders are not same.