, XII- MATHS Matrices - Results
Matrix.
A matrix is an ordered rectangular array of numbers (or functions). The numbers (or functions) are
called the elements or the entries of the matrix.
The horizontal lines of elements are said to constitute rows of the matrix and the
vertical lines of elements are said to constitute columns of the matrix.
Order of a Matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix
(read as an m by n matrix).
In general, an m × n matrix has the following rectangular array :
𝑎!! 𝑎!" 𝑎!# … . . 𝑎!$
𝐴 = [𝑎𝑖𝑗]𝑚 × 𝑛 = , 𝑎"! .
⋱ . . ⋮ 1 , 1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛 𝑖, 𝑗 ∈ 𝑵.
𝑎%! .. 𝑎%$
The element, aij is an element lying in the ith row and jh column and is known as the (i, j)th
element of A. The number of elements in an m × n matrix is mn.
Types of Matrices
§ Row matrix : It has only one row. Order is of the form 1 × n
§ Column matrix : It has only one column. Order is of the form m × 1
§ Square matrix : The number of rows are equal to the number of columns. Thus, an m × n
matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’.
For a square matrix, the leading diagonal or diagonal is the diagonal where the
elements have the same number of row and column. i.e; the elements for which i = j.
𝑎 𝑏 𝑐
!𝑏 ℎ 𝑑 ( The elements a, h , g form the diagonal of the matrix.
𝑐 𝑑 𝑔
§ Diagonal matrix : A square matrix B = [𝑏&' ]$×$ is said to be a diagonal matrix if it’s all
non-diagonal elements are zero.
that is a matrix B = [𝑏&' ]$×$ is said to be a diagonal matrix if 𝒃𝒊𝒋 = 0, when 𝒊 ≠ 𝒋.
§ Scalar matrix A diagonal matrix is said to be a scalar matrix if it’s diagonal elements are
equal.
𝟎, 𝒘𝒉𝒆𝒏 𝒊 ≠ 𝒋
i.e; a square matrix B = [𝑏&' ]$×$ is said to be a scalar matrix if 𝒃𝒊𝒋 = < for
𝒌, 𝒘𝒉𝒆𝒏 𝒊 = 𝒋
some constant k.
§ Identity matrix A square matrix in which all elements in the diagonal are all 1 and rest are
all zeroes is called an identity matrix.
𝟎, 𝒘𝒉𝒆𝒏 𝒊 ≠ 𝒋
i.e; the square matrix A = [𝑎&' ]$×$ is an identity matrix, if 𝒂𝒊𝒋 = < It is
𝟏, 𝒘𝒉𝒆𝒏 𝒊 = 𝒋
denoted by I.
§ Zero/Null matrix : A matrix is said to be zero matrix if all its elements are zeroes, denoted
by O.
Equality of Matrices.
Two matrices A = [aij] and B = [bij] are said to be equal if
(a) they are of the same order, and
(b) each element of A is equal to the corresponding element of B, that is, aij = bij for all i and j.
© Dineshmaths 0509312916 XII Mathematics
Matrix.
A matrix is an ordered rectangular array of numbers (or functions). The numbers (or functions) are
called the elements or the entries of the matrix.
The horizontal lines of elements are said to constitute rows of the matrix and the
vertical lines of elements are said to constitute columns of the matrix.
Order of a Matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix
(read as an m by n matrix).
In general, an m × n matrix has the following rectangular array :
𝑎!! 𝑎!" 𝑎!# … . . 𝑎!$
𝐴 = [𝑎𝑖𝑗]𝑚 × 𝑛 = , 𝑎"! .
⋱ . . ⋮ 1 , 1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛 𝑖, 𝑗 ∈ 𝑵.
𝑎%! .. 𝑎%$
The element, aij is an element lying in the ith row and jh column and is known as the (i, j)th
element of A. The number of elements in an m × n matrix is mn.
Types of Matrices
§ Row matrix : It has only one row. Order is of the form 1 × n
§ Column matrix : It has only one column. Order is of the form m × 1
§ Square matrix : The number of rows are equal to the number of columns. Thus, an m × n
matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’.
For a square matrix, the leading diagonal or diagonal is the diagonal where the
elements have the same number of row and column. i.e; the elements for which i = j.
𝑎 𝑏 𝑐
!𝑏 ℎ 𝑑 ( The elements a, h , g form the diagonal of the matrix.
𝑐 𝑑 𝑔
§ Diagonal matrix : A square matrix B = [𝑏&' ]$×$ is said to be a diagonal matrix if it’s all
non-diagonal elements are zero.
that is a matrix B = [𝑏&' ]$×$ is said to be a diagonal matrix if 𝒃𝒊𝒋 = 0, when 𝒊 ≠ 𝒋.
§ Scalar matrix A diagonal matrix is said to be a scalar matrix if it’s diagonal elements are
equal.
𝟎, 𝒘𝒉𝒆𝒏 𝒊 ≠ 𝒋
i.e; a square matrix B = [𝑏&' ]$×$ is said to be a scalar matrix if 𝒃𝒊𝒋 = < for
𝒌, 𝒘𝒉𝒆𝒏 𝒊 = 𝒋
some constant k.
§ Identity matrix A square matrix in which all elements in the diagonal are all 1 and rest are
all zeroes is called an identity matrix.
𝟎, 𝒘𝒉𝒆𝒏 𝒊 ≠ 𝒋
i.e; the square matrix A = [𝑎&' ]$×$ is an identity matrix, if 𝒂𝒊𝒋 = < It is
𝟏, 𝒘𝒉𝒆𝒏 𝒊 = 𝒋
denoted by I.
§ Zero/Null matrix : A matrix is said to be zero matrix if all its elements are zeroes, denoted
by O.
Equality of Matrices.
Two matrices A = [aij] and B = [bij] are said to be equal if
(a) they are of the same order, and
(b) each element of A is equal to the corresponding element of B, that is, aij = bij for all i and j.
© Dineshmaths 0509312916 XII Mathematics
