3.2 Types of Matrices

3.2 Types of Matrices

(i) Row Matrix: A matrix having only one row is called a row matrix.

For example: $A = [0 \quad 1 \quad 4], B = [a \quad b], C = [a]$ are row matrices of order $1$-by-$3$, $1$-by-$2$ and $1$-by-$1$ respectively.

(ii) Column Matrix: A matrix having only one column is called a column matrix.

For example: $A = \begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix}, B = \begin{bmatrix} a \\ b \end{bmatrix}, C = [2]$ are column matrices of order $3$-by-$1$, $2$-by-$1$ and $1$-by-$1$ respectively.

(iii) Zero matrix or Null matrix

If all the entries in a matrix are zero, it is called zero matrix or null matrix. It is represented by $O$.

For example: $[0], \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ are zero matrices of order $1$-by-$1$, $2$-by-$2$ and $2$-by-$3$ respectively.

(iv) Square Matrix: A matrix in which number of rows is equal to the number of columns is called a square matrix. A matrix of order $n$-by-$n$ is often referred to as a square matrix of order $n$.

For example: $A = \begin{bmatrix} a & b & c \\ d & c & f \\ g & h & l \end{bmatrix}$ is a square matrix of order $3$.

In a square matrix $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$

$a_{11}, a_{22}, a_{33}$ are called entries of principal diagonal or main diagonal or leading diagonal.

(v) Rectangular Matrix: If the number of rows and number of columns in a matrix are not equal, then it is called a rectangular matrix.

For example: $A = [a \quad b], B = \begin{bmatrix} a \\ b \end{bmatrix}, C = \begin{bmatrix} 4 & 0 & 7 \\ 8 & 3 & 1 \end{bmatrix}$ are rectangular matrices.

(vi) Diagonal Matrix: A square matrix is called a diagonal matrix if at least any one of the elements of its main diagonal is non-zero and non-diagonal elements are zero.

For example: $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}, B = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ are diagonal matrices of order $3$-by-$3$ and $2$-by-$2$ respectively.

(vii) Scalar Matrix: A diagonal matrix is called a scalar matrix if all the entries in the main diagonal are the same and non-zero.

For example: $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}, B = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}, C = [\sqrt{3}]$ are scalar matrices of order $3$-by-$3$, $2$-by-$2$ and $1$-by-$1$ respectively.

(viii) Identity Matrix or Unit Matrix: A scalar matrix is called an identity or unit matrix if all of its main diagonal entries are $1$.

For example: $P = [1], Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ are identity matrices of order $1$-by-$1$, $2$-by-$2$ and $3$-by-$3$ respectively. We represent these matrices as:

$I_1 = [1], I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

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Remember!
A scalar matrix and identity matrix are also diagonal matrices but all diagonal matrices are not scalar or identity matrices.

(ix) Transpose of a Matrix: A matrix obtained from a given matrix by interchanging the rows and columns is called the transpose of the given matrix. If $A$ is a matrix, then its transpose is represented as $A^t$.

For example: $A = \begin{bmatrix} 1 & 2 & 4 \\ 8 & 0 & 5 \end{bmatrix} \Rightarrow A^t = \begin{bmatrix} 1 & 8 \\ 2 & 0 \\ 4 & 5 \end{bmatrix}$

(x) Symmetric Matrix: A square matrix $A$ is called symmetric if it is equal to its transpose. i.e., $A^t = A$.

For example: If $A = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$, then $A^t = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} = A$

Thus, $A$ is a symmetric matrix.

(xi) Skew-Symmetric Matrix: A square matrix $A$ is called skew-symmetric if $A^t = -A$.

For example: If $B = \begin{bmatrix} 0 & -5 \\ 5 & 0 \end{bmatrix}$, then $B^t = \begin{bmatrix} 0 & 5 \\ -5 & 0 \end{bmatrix} = -B$

Thus, $B$ is a skew-symmetric matrix.

(xii) Negative of a Matrix: The negative of a matrix $A$ is obtained by multiplying each of its elements by $-1$. It is denoted by $-A$.

For example: If $A = [3 \quad -6]$, then

$-A = [-3 \quad 6]$