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Matrices
A rectangular array of mn numbers in the form of m horizontal lines (called IN THIS CHAPTER ....
rows) and n vertical lines (called columns), is called a matrix of order m ´ n.
Matrix
This type of array is enclosed by [ ] or ( )
Types of Matrices
Each of mn numbers of a matrix is known as element of a matrix. A matrix is
generally denoted by A, B, C , ... etc., and its element is denoted by aij , where Equality of Two Matrices
aij belongs to the ith row and jth column and is called ( i , j)th element of the Algebra of Matrices
matrix A = [aij ] . Trace of a Matrix
An m ´ n matrix is usually written as Transpose of a Matrix
é a11 a12 K a1n ù
Special Types of Matrices
êa a22 K a2n ú
A = ê 21 ú = [aij ]m ´ n Elementary Row Transformations
ê M M M ú
Elementary Matrix
êa ú
ë m1 am 2 K amn û
é3 2 7ù
e.g. A = ê 5 -4 6 ú is a matrix of order 3 ´ 3.
ê ú
êë 4 8 -12úû
Types of Matrices
Row Matrix
A matrix which has only one row and any number of columns, is called a row
matrix. e.g. A = [ 27 85 1 4 ] 1 ´ 4 is a row matrix.
Column Matrix
A matrix is said to be a column matrix, if it has only one column and any
number of rows.
é 1ù
e.g. A = ê 2ú is a column matrix.
ê ú
êë úû3 ´ 1
3
, 532 JEE Main Mathematics
Rectangular Matrix Scalar Matrix
A matrix in which number of rows is not equal to the A square matrix A = [aij ] is said to be scalar matrix, if
number of columns or vice-versa is called a rectangular (a) aij = 0, " i ¹ j
matrix. (b) aij = k, " i = j, where k ¹ 0
é 1 2 3ù
e.g. A = ê ú is a rectangular matrix of order 2 ´ 3. i.e. a diagonal matrix is said to be a scalar matrix, if the
ë 4 5 6û elements of principle diagonal are same.
Square Matrix é 5 0 0ù
A matrix in which number of rows is equal to the number e.g. A = ê 0 5 0ú is a scalar matrix.
ê ú
of columns, is called a square matrix. The elements aij of êë 0 0 5úû
a square matrix A = [aij ]m ´ m for which i = j i.e, the
elements a11 , a22 , . . . , amm are called the diagonal Unit Matrix or Identity Matrix
elements and the line along which they lie is called the A square matrix A = [aij ] is said to be a unit matrix or
principal diagonal or leading diagonal of the matrix. identity matrix, if
é 1 2 3ù (a) aij = 0, " i ¹ j (b) aij = 1, " i = j
e.g. A = ê 3 2 1ú i.e. A diagonal matrix, whose elements of principle
ê ú
êë 2 3 1 úû3 ´ 3 diagonal are equal to 1 and all remaining elements
are zero, is known as unit or identity matrix. It is
is a square matrix of order 3 in which diagonal elements denoted by I.
are 1, 2, 1. é 1 0 0ù
e.g. I = ê 0 1 0ú is a unit matrix of order 3.
Null Matrix ê ú
A matrix of order m ´ n whose all elements are zero, is êë 0 0 1 úû
called a null matrix of order m ´ n.
Upper Triangular Matrix
It is denoted by O.
A square matrix A = [aij ] is known as upper triangular
é 0 0ù é0 0 0ù matrix, if
e.g. ê 0 0ú and ê 0 0
ë û ë 0úû aij = 0, " i > j.
are two null matrices of order 2 ´ 2 and 2 ´ 3, respectively. é 1 4 5ù
e.g. A = ê 0 2 6ú is an upper triangular matrix.
Diagonal Matrix ê ú
A square matrix is called a diagonal matrix, if all its êë 0 0 3úû
non-diagonal elements are zero and diagonal elements
are not all equal.
Lower Triangular Matrix
If d1 , d2 , d3 , ... , dn are elements of principal diagonal of a A square matrix A = [aij ] is known as lower triangular
diagonal matrix of order n ´ n, then matrix is denoted as matrix, if aij = 0, " i < j.
diag [d1 , d2 , ... , dn ]. é 1 0 0ù
é a 0 0ù e.g. A = ê 4 2 0ú is a lower triangular matrix.
ê ú
e.g. A = ê 0 b 0ú is a diagonal matrix which is denoted êë 5 6 3úû
ê ú
êë 0 0 c úû Submatrices of a Matrix
by A = diag [a , b, c]. A matrix B obtained by deleting the row (s) or Column (s)
Note The number of zeroes in a diagonal matrix is given by or both of a matrix A is said to be a submatrix of A.
n 2 - n, where n is an order of the matrix. i.e. The matrix B constituted by the array of elements,
which are left after deleting some rows or columns or
Triple Diagonal Matrix both of matrix A is called submatrix of A.
A square matrix A is said to be a triple diagonal matrix, (a) Principle Submatrix A square submatrix B of a
if all its elements are zero except possibly for those lying square matrix A is called a principle submatrix, if
on the principle diagonal, the diagonal immediately the diagonal elements of B are aslo diagonal
above as well as below the principle diagonal. elements of A.
é 1 -1 0 0ù (b) Leading Submatrix A principle square submatrix
é 5 -3 0 ù ê ú B is said to be a leading submatrix of a square
e.g. ê -3 4 -3ú and ê -1 2 -1 0ú
ê ú matrix A if it is obtained by deleting only some of
ê0 1 2 3ú
the last rows and the corresponding columns such
êë 0 0 4 úû ê 0 0 4 5ú
ë û that the leading elements (i.e. a11) is not lost.
Matrices
A rectangular array of mn numbers in the form of m horizontal lines (called IN THIS CHAPTER ....
rows) and n vertical lines (called columns), is called a matrix of order m ´ n.
Matrix
This type of array is enclosed by [ ] or ( )
Types of Matrices
Each of mn numbers of a matrix is known as element of a matrix. A matrix is
generally denoted by A, B, C , ... etc., and its element is denoted by aij , where Equality of Two Matrices
aij belongs to the ith row and jth column and is called ( i , j)th element of the Algebra of Matrices
matrix A = [aij ] . Trace of a Matrix
An m ´ n matrix is usually written as Transpose of a Matrix
é a11 a12 K a1n ù
Special Types of Matrices
êa a22 K a2n ú
A = ê 21 ú = [aij ]m ´ n Elementary Row Transformations
ê M M M ú
Elementary Matrix
êa ú
ë m1 am 2 K amn û
é3 2 7ù
e.g. A = ê 5 -4 6 ú is a matrix of order 3 ´ 3.
ê ú
êë 4 8 -12úû
Types of Matrices
Row Matrix
A matrix which has only one row and any number of columns, is called a row
matrix. e.g. A = [ 27 85 1 4 ] 1 ´ 4 is a row matrix.
Column Matrix
A matrix is said to be a column matrix, if it has only one column and any
number of rows.
é 1ù
e.g. A = ê 2ú is a column matrix.
ê ú
êë úû3 ´ 1
3
, 532 JEE Main Mathematics
Rectangular Matrix Scalar Matrix
A matrix in which number of rows is not equal to the A square matrix A = [aij ] is said to be scalar matrix, if
number of columns or vice-versa is called a rectangular (a) aij = 0, " i ¹ j
matrix. (b) aij = k, " i = j, where k ¹ 0
é 1 2 3ù
e.g. A = ê ú is a rectangular matrix of order 2 ´ 3. i.e. a diagonal matrix is said to be a scalar matrix, if the
ë 4 5 6û elements of principle diagonal are same.
Square Matrix é 5 0 0ù
A matrix in which number of rows is equal to the number e.g. A = ê 0 5 0ú is a scalar matrix.
ê ú
of columns, is called a square matrix. The elements aij of êë 0 0 5úû
a square matrix A = [aij ]m ´ m for which i = j i.e, the
elements a11 , a22 , . . . , amm are called the diagonal Unit Matrix or Identity Matrix
elements and the line along which they lie is called the A square matrix A = [aij ] is said to be a unit matrix or
principal diagonal or leading diagonal of the matrix. identity matrix, if
é 1 2 3ù (a) aij = 0, " i ¹ j (b) aij = 1, " i = j
e.g. A = ê 3 2 1ú i.e. A diagonal matrix, whose elements of principle
ê ú
êë 2 3 1 úû3 ´ 3 diagonal are equal to 1 and all remaining elements
are zero, is known as unit or identity matrix. It is
is a square matrix of order 3 in which diagonal elements denoted by I.
are 1, 2, 1. é 1 0 0ù
e.g. I = ê 0 1 0ú is a unit matrix of order 3.
Null Matrix ê ú
A matrix of order m ´ n whose all elements are zero, is êë 0 0 1 úû
called a null matrix of order m ´ n.
Upper Triangular Matrix
It is denoted by O.
A square matrix A = [aij ] is known as upper triangular
é 0 0ù é0 0 0ù matrix, if
e.g. ê 0 0ú and ê 0 0
ë û ë 0úû aij = 0, " i > j.
are two null matrices of order 2 ´ 2 and 2 ´ 3, respectively. é 1 4 5ù
e.g. A = ê 0 2 6ú is an upper triangular matrix.
Diagonal Matrix ê ú
A square matrix is called a diagonal matrix, if all its êë 0 0 3úû
non-diagonal elements are zero and diagonal elements
are not all equal.
Lower Triangular Matrix
If d1 , d2 , d3 , ... , dn are elements of principal diagonal of a A square matrix A = [aij ] is known as lower triangular
diagonal matrix of order n ´ n, then matrix is denoted as matrix, if aij = 0, " i < j.
diag [d1 , d2 , ... , dn ]. é 1 0 0ù
é a 0 0ù e.g. A = ê 4 2 0ú is a lower triangular matrix.
ê ú
e.g. A = ê 0 b 0ú is a diagonal matrix which is denoted êë 5 6 3úû
ê ú
êë 0 0 c úû Submatrices of a Matrix
by A = diag [a , b, c]. A matrix B obtained by deleting the row (s) or Column (s)
Note The number of zeroes in a diagonal matrix is given by or both of a matrix A is said to be a submatrix of A.
n 2 - n, where n is an order of the matrix. i.e. The matrix B constituted by the array of elements,
which are left after deleting some rows or columns or
Triple Diagonal Matrix both of matrix A is called submatrix of A.
A square matrix A is said to be a triple diagonal matrix, (a) Principle Submatrix A square submatrix B of a
if all its elements are zero except possibly for those lying square matrix A is called a principle submatrix, if
on the principle diagonal, the diagonal immediately the diagonal elements of B are aslo diagonal
above as well as below the principle diagonal. elements of A.
é 1 -1 0 0ù (b) Leading Submatrix A principle square submatrix
é 5 -3 0 ù ê ú B is said to be a leading submatrix of a square
e.g. ê -3 4 -3ú and ê -1 2 -1 0ú
ê ú matrix A if it is obtained by deleting only some of
ê0 1 2 3ú
the last rows and the corresponding columns such
êë 0 0 4 úû ê 0 0 4 5ú
ë û that the leading elements (i.e. a11) is not lost.
