Matrix

A rectangular array of numbers arranged in horizontal and vertical enclosed between round (or square) brackets is called the matrix. Horizontal lines are called rows and vertical lines are called column of the matrix.

Order of a matrix

The order or the size of the matrix is given by the number of rows followed by the number of columns. If a matrix contains m rows and n columns, then it is of order m×n, read as m by n.

\(\begin{pmatrix} 2&6\\4&3\end{pmatrix}\) is a 2x2 matrix and\(\begin{pmatrix} 2&3&4\\5&2&9\end{pmatrix}\) is a 3x3 matrix.

A matrix of order m×n has mn elements.

Notation

The matrices are usually denoted by a capital letter such as A, B, C,....etc. The elements are denoted by the corresponding small letters along with two suffixes. The first suffix indicates the number of rows and the latter one indicates the number of columns in which the element appears.

a_ij is the element of a matrix A in the i^th row and j^th column.

a₂₃ is the element of a matrix A in the 2^nd row and 3^rd column.

Thus, a matrix of order m×n may be written as

A =(a_ij)m×n.

If A is a 3x3 matrix, then it may be written as

A=\(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\)

Types of Matrices

1. Row matrix
   A matrix having only one-row ia called a matrix. For example (1 2 4) is a row matrix of order 1x3. Similarly{1 5}, {3-1 7 9} are matrices of order 1x2 and 1x4 respectively.
   
   
2. Column matrix
   A matrix having only one column is called a column matrix. For example, the matrix\(\begin{bmatrix} {1}\\{2} \end{bmatrix}\) ,\(\begin{bmatrix} {3}\\{2}\\{6} \end{bmatrix}\) and\(\begin{bmatrix} {2}\\{9}\\{3}\\{7} \end{bmatrix}\) are column matrix of order 2x1, 3x1 and 4x1 respectively.
   
   
3. Square matrix
   A matrix having the same number of rows and column is called a square matrix. For example, the matrices \(\begin{bmatrix} 1&2\\3&4\end{bmatrix}\) \(\begin{bmatrix} 1&0&2\\3&1&4\\2&0&5\end{bmatrix}\) are square matrices of order 2x2 & 3x3 respectively. A square matrix having order nxn is called a square matrix of order n. Thus the matrices given are of order 2x3 respectively.
   
   
4. Rectangular Matrix
   A matrix in which the number of rows is not equal to the number of columns is called a rectangular matrix. For example, the matrices\(\begin{bmatrix} 1&2&3\\4&0&5\end{bmatrix}\) and\(\begin{bmatrix} 1&7\\2&0\\4&9\end{bmatrix}\) are rectangular matrices of order 2x3 and 3x2 respectively.
   
   
5. Zero Matrix or Null Matrix
   If each element of a matrix is zero, then the matrix is called zero matrix or null matrix. For example, the matrices, \(\begin{bmatrix} 0&0\\0&0\end{bmatrix}\) , \(\begin{bmatrix} 0&0&0\\0&0&0\end{bmatrix}\) & \(\begin{bmatrix} 0&0&0\\0&0&0\\0&0&0\end{bmatrix}\) are zero matrices of order 2x2, 2x3 and 3x3 respectivelty. Zero matrix is denoted by letter 0.
   
   
6. Diagonal Matrix
   In a square matrix, the diagonal from the left top to the right bottom is called Principal diagonal elements. A square matrix having all non-diagonal elements zero is called a diagonal matrix. For example, the square matrix,\(\begin{bmatrix} 1&0\\0&2\end{bmatrix}\) &\(\begin{bmatrix} 1&0&0\\0&2&0\\0&0&3\end{bmatrix}\) are diagonal matrices of order 2&3 repectively.
   
   
7. Scalar Matrix
   If all the diagonal elements of a diagonal matrix are same, then the matrix is called a scalar matrix. For example, the matrices. \(\begin{bmatrix} 8&0\\0&8\end{bmatrix}\) & \(\begin{bmatrix} -2&0&0\\0&-2&0\\0&0&-2\end{bmatrix}\) are scalar matrices of order 2and 3 respectively. Note that scalar matrix is a particuaar case of a diagonal matrix.
   
   
8. Unit Matrix or IdentityMatrix
   If all the diagonal elements of a diagonal are unity, then the matrix is called on identity or unit matrix. For example, the matrices, \(\begin{bmatrix} 1&0\\0&1\end{bmatrix}\) and\(\begin{bmatrix} 1&0&0\\0&1&0\\0&0&1\end{bmatrix}\) are unity matrices of order 2&3. An identity Matrix is denoted by the letter I. Note that a unit is a particular case of a scalar matrix.
   
   
9. Triangular Matrix
   A square matrix having all the elements either above the leading diagonal are zero or below the leading diagonal zero is called the triangular matrix. \begin{bmatrix} 2&0\\-1&3\end{bmatrix}.
   
   
10. Equal Matrices

Two matrices A&B are said to be equal matrices if A and B are of some order i.e. number of rows in A = number of rows in B and number of columns in A = number of columns in B, and their corresponding elements are equal i.e. the entries of A and B in some position are equal. Otherwise, the matrices are said to be unequal. If A and B equal matrices, then we write A=B. Otherwise, we write A ≠B.