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# Matrices - Determinants Class 12 Notes | EduRev

## Class 12: Matrices - Determinants Class 12 Notes | EduRev

The document Matrices - Determinants Class 12 Notes | EduRev is a part of Class 12 category.
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``` Page 1

Matrices and Determinants
1. Matrices
1.1. Basic concepts
A matrix is a rectangular array of (real or complex) numbers:
A =
2
6
6
6
4
a
11
a
12
 a
1m
a
21
a
22
 a
2m
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nm
3
7
7
7
5
The numbers in the matrix are called its entries.
The size of a matrix is described by the number of its rows and its columns. A
has n rows and m columns, thus it is an nm (or: n by m) matrix.
Matrices A and B are equal if a
ij
= b
ij
for any i and j, and A and B are of
the same size.
A matrix with just one column is a column vector:
b =
2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
A matrix with just one row is a row vector:
c =

c
1
c
2
 c
m

A square matrix has an equal number of rows and columns:
S =
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nn
3
7
7
7
5
A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes:
0 =
2
6
6
6
4
0 0  0
0 0  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  0
3
7
7
7
5
1
Page 2

Matrices and Determinants
1. Matrices
1.1. Basic concepts
A matrix is a rectangular array of (real or complex) numbers:
A =
2
6
6
6
4
a
11
a
12
 a
1m
a
21
a
22
 a
2m
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nm
3
7
7
7
5
The numbers in the matrix are called its entries.
The size of a matrix is described by the number of its rows and its columns. A
has n rows and m columns, thus it is an nm (or: n by m) matrix.
Matrices A and B are equal if a
ij
= b
ij
for any i and j, and A and B are of
the same size.
A matrix with just one column is a column vector:
b =
2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
A matrix with just one row is a row vector:
c =

c
1
c
2
 c
m

A square matrix has an equal number of rows and columns:
S =
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nn
3
7
7
7
5
A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes:
0 =
2
6
6
6
4
0 0  0
0 0  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  0
3
7
7
7
5
1
A square matrix is called diagonal if all of its entries, apart from the leading
diagonal(top left to lower right), are zeroes:
D =
2
6
6
6
4
a
11
0  0
0 a
22
 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  a
nn
3
7
7
7
5
A diagonal matrix is called unit matrix if all of its entries in the leading diagonal
are 1's (and all of its other entries are zeroes):
I =
2
6
6
6
4
1 0  0
0 1  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  1
3
7
7
7
5
A square matrix is symmetrical if a
ij
=a
ji
for any i and j:
F =
2
6
6
6
4
a
11
a
12
 a
1n
a
12
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 a
nn
3
7
7
7
5
A square matrix is skew-symmetrical if a
ij
=a
ji
for any i and j. In this case
any entry in the leading diagonal needs to be 0:
G =
2
6
6
6
4
0 a
12
 a
1n
a
12
0  a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 0
3
7
7
7
5
2
Page 3

Matrices and Determinants
1. Matrices
1.1. Basic concepts
A matrix is a rectangular array of (real or complex) numbers:
A =
2
6
6
6
4
a
11
a
12
 a
1m
a
21
a
22
 a
2m
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nm
3
7
7
7
5
The numbers in the matrix are called its entries.
The size of a matrix is described by the number of its rows and its columns. A
has n rows and m columns, thus it is an nm (or: n by m) matrix.
Matrices A and B are equal if a
ij
= b
ij
for any i and j, and A and B are of
the same size.
A matrix with just one column is a column vector:
b =
2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
A matrix with just one row is a row vector:
c =

c
1
c
2
 c
m

A square matrix has an equal number of rows and columns:
S =
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nn
3
7
7
7
5
A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes:
0 =
2
6
6
6
4
0 0  0
0 0  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  0
3
7
7
7
5
1
A square matrix is called diagonal if all of its entries, apart from the leading
diagonal(top left to lower right), are zeroes:
D =
2
6
6
6
4
a
11
0  0
0 a
22
 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  a
nn
3
7
7
7
5
A diagonal matrix is called unit matrix if all of its entries in the leading diagonal
are 1's (and all of its other entries are zeroes):
I =
2
6
6
6
4
1 0  0
0 1  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  1
3
7
7
7
5
A square matrix is symmetrical if a
ij
=a
ji
for any i and j:
F =
2
6
6
6
4
a
11
a
12
 a
1n
a
12
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 a
nn
3
7
7
7
5
A square matrix is skew-symmetrical if a
ij
=a
ji
for any i and j. In this case
any entry in the leading diagonal needs to be 0:
G =
2
6
6
6
4
0 a
12
 a
1n
a
12
0  a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 0
3
7
7
7
5
2
1.2. Operations on Matrices
1.2.1. The Transpose of a Matrix
The transpose A
T
of a matrix A is obtained by interchanging the rows and
columns of A.
Property:
(A
T
)
T
=A.
If two matrices are of the same size, they can be added by summing the corres-
pondig entries. (Subtraction can be dened similarly.)
Properties:
(a) A +B =B +A (addition is commutative);
(b) (A +B) +C =A + (B +C) (addition is associative);
(c) (A +B)
T
=A
T
+B
T
.
1.2.3. Multiplication by a Scalar
If a
ij
is a typical entry of A, then A is a matrix, whose corresponding entry
is a
ij
.
Properties:
(a) ()A =(A);
(b) ( +)A =A +A;
(c) (A +B) =A +B.
1.2.4. Multiplication of Matrices
(a) A row vector and a column vector with the same number of entries can
be multiplied as the scalar product is dened:

a
1
a
2
 a
n


2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
=a
1
b
1
+a
2
b
2
+::: +a
n
b
n
(b) The denition can be extended to cover the productAB of any two matri-
ces, provided that the number of columns in A is the same as the number
of rows in B.
3
Page 4

Matrices and Determinants
1. Matrices
1.1. Basic concepts
A matrix is a rectangular array of (real or complex) numbers:
A =
2
6
6
6
4
a
11
a
12
 a
1m
a
21
a
22
 a
2m
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nm
3
7
7
7
5
The numbers in the matrix are called its entries.
The size of a matrix is described by the number of its rows and its columns. A
has n rows and m columns, thus it is an nm (or: n by m) matrix.
Matrices A and B are equal if a
ij
= b
ij
for any i and j, and A and B are of
the same size.
A matrix with just one column is a column vector:
b =
2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
A matrix with just one row is a row vector:
c =

c
1
c
2
 c
m

A square matrix has an equal number of rows and columns:
S =
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nn
3
7
7
7
5
A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes:
0 =
2
6
6
6
4
0 0  0
0 0  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  0
3
7
7
7
5
1
A square matrix is called diagonal if all of its entries, apart from the leading
diagonal(top left to lower right), are zeroes:
D =
2
6
6
6
4
a
11
0  0
0 a
22
 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  a
nn
3
7
7
7
5
A diagonal matrix is called unit matrix if all of its entries in the leading diagonal
are 1's (and all of its other entries are zeroes):
I =
2
6
6
6
4
1 0  0
0 1  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  1
3
7
7
7
5
A square matrix is symmetrical if a
ij
=a
ji
for any i and j:
F =
2
6
6
6
4
a
11
a
12
 a
1n
a
12
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 a
nn
3
7
7
7
5
A square matrix is skew-symmetrical if a
ij
=a
ji
for any i and j. In this case
any entry in the leading diagonal needs to be 0:
G =
2
6
6
6
4
0 a
12
 a
1n
a
12
0  a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 0
3
7
7
7
5
2
1.2. Operations on Matrices
1.2.1. The Transpose of a Matrix
The transpose A
T
of a matrix A is obtained by interchanging the rows and
columns of A.
Property:
(A
T
)
T
=A.
If two matrices are of the same size, they can be added by summing the corres-
pondig entries. (Subtraction can be dened similarly.)
Properties:
(a) A +B =B +A (addition is commutative);
(b) (A +B) +C =A + (B +C) (addition is associative);
(c) (A +B)
T
=A
T
+B
T
.
1.2.3. Multiplication by a Scalar
If a
ij
is a typical entry of A, then A is a matrix, whose corresponding entry
is a
ij
.
Properties:
(a) ()A =(A);
(b) ( +)A =A +A;
(c) (A +B) =A +B.
1.2.4. Multiplication of Matrices
(a) A row vector and a column vector with the same number of entries can
be multiplied as the scalar product is dened:

a
1
a
2
 a
n


2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
=a
1
b
1
+a
2
b
2
+::: +a
n
b
n
(b) The denition can be extended to cover the productAB of any two matri-
ces, provided that the number of columns in A is the same as the number
of rows in B.
3
IfA is of sizenm andB is of sizemp then the product AB is matrix
C of sizenp, whosec
ij
entry is the scalar product of the i-th row of A
and the j-th column of B.
AB =
2
6
4
b
11
 b
1j
 b
1p
.
.
.
.
.
.
.
.
.
b
m1
 b
mj
 b
mp
3
7
5
2
6
6
6
6
6
6
4
a
11
 a
1m
.
.
.
.
.
.
a
i1
 a
im
.
.
.
.
.
.
a
n1
 a
nm
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
c
11
 c
1j
 c
1p
.
.
.
.
.
.
.
.
.
.
.
.
c
i1
 c
ij
 c
ip
.
.
.
.
.
.
.
.
.
.
.
.
c
n1
 c
nj
 c
np
3
7
7
7
7
7
7
5
Properties:
(a) If AB exists, BA does not necessarily exist. If both AB and BA exist,
then in general AB6=BA (multiplication is not commutative);
(b) Taking the product ofA and the unit matrix (I) of suitable size, the result
is A:
A
nm
I
m
=A
nm
=I
n
A
nm
;
(c) Multiplcation is associative:
(AB)C =A (BC);
(d) The distributive laws are valid:
(A +B)C =AC +BC and A (B +C) =AB +AC;
(e) Transposing reverses the order of products:
(AB)
T
=B
T
A
T
:
4
Page 5

Matrices and Determinants
1. Matrices
1.1. Basic concepts
A matrix is a rectangular array of (real or complex) numbers:
A =
2
6
6
6
4
a
11
a
12
 a
1m
a
21
a
22
 a
2m
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nm
3
7
7
7
5
The numbers in the matrix are called its entries.
The size of a matrix is described by the number of its rows and its columns. A
has n rows and m columns, thus it is an nm (or: n by m) matrix.
Matrices A and B are equal if a
ij
= b
ij
for any i and j, and A and B are of
the same size.
A matrix with just one column is a column vector:
b =
2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
A matrix with just one row is a row vector:
c =

c
1
c
2
 c
m

A square matrix has an equal number of rows and columns:
S =
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
 a
nn
3
7
7
7
5
A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes:
0 =
2
6
6
6
4
0 0  0
0 0  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  0
3
7
7
7
5
1
A square matrix is called diagonal if all of its entries, apart from the leading
diagonal(top left to lower right), are zeroes:
D =
2
6
6
6
4
a
11
0  0
0 a
22
 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  a
nn
3
7
7
7
5
A diagonal matrix is called unit matrix if all of its entries in the leading diagonal
are 1's (and all of its other entries are zeroes):
I =
2
6
6
6
4
1 0  0
0 1  0
.
.
.
.
.
.
.
.
.
.
.
.
0 0  1
3
7
7
7
5
A square matrix is symmetrical if a
ij
=a
ji
for any i and j:
F =
2
6
6
6
4
a
11
a
12
 a
1n
a
12
a
22
 a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 a
nn
3
7
7
7
5
A square matrix is skew-symmetrical if a
ij
=a
ji
for any i and j. In this case
any entry in the leading diagonal needs to be 0:
G =
2
6
6
6
4
0 a
12
 a
1n
a
12
0  a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
1n
a
2n
 0
3
7
7
7
5
2
1.2. Operations on Matrices
1.2.1. The Transpose of a Matrix
The transpose A
T
of a matrix A is obtained by interchanging the rows and
columns of A.
Property:
(A
T
)
T
=A.
If two matrices are of the same size, they can be added by summing the corres-
pondig entries. (Subtraction can be dened similarly.)
Properties:
(a) A +B =B +A (addition is commutative);
(b) (A +B) +C =A + (B +C) (addition is associative);
(c) (A +B)
T
=A
T
+B
T
.
1.2.3. Multiplication by a Scalar
If a
ij
is a typical entry of A, then A is a matrix, whose corresponding entry
is a
ij
.
Properties:
(a) ()A =(A);
(b) ( +)A =A +A;
(c) (A +B) =A +B.
1.2.4. Multiplication of Matrices
(a) A row vector and a column vector with the same number of entries can
be multiplied as the scalar product is dened:

a
1
a
2
 a
n


2
6
6
6
4
b
1
b
2
.
.
.
b
n
3
7
7
7
5
=a
1
b
1
+a
2
b
2
+::: +a
n
b
n
(b) The denition can be extended to cover the productAB of any two matri-
ces, provided that the number of columns in A is the same as the number
of rows in B.
3
IfA is of sizenm andB is of sizemp then the product AB is matrix
C of sizenp, whosec
ij
entry is the scalar product of the i-th row of A
and the j-th column of B.
AB =
2
6
4
b
11
 b
1j
 b
1p
.
.
.
.
.
.
.
.
.
b
m1
 b
mj
 b
mp
3
7
5
2
6
6
6
6
6
6
4
a
11
 a
1m
.
.
.
.
.
.
a
i1
 a
im
.
.
.
.
.
.
a
n1
 a
nm
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
c
11
 c
1j
 c
1p
.
.
.
.
.
.
.
.
.
.
.
.
c
i1
 c
ij
 c
ip
.
.
.
.
.
.
.
.
.
.
.
.
c
n1
 c
nj
 c
np
3
7
7
7
7
7
7
5
Properties:
(a) If AB exists, BA does not necessarily exist. If both AB and BA exist,
then in general AB6=BA (multiplication is not commutative);
(b) Taking the product ofA and the unit matrix (I) of suitable size, the result
is A:
A
nm
I
m
=A
nm
=I
n
A
nm
;
(c) Multiplcation is associative:
(AB)C =A (BC);
(d) The distributive laws are valid:
(A +B)C =AC +BC and A (B +C) =AB +AC;
(e) Transposing reverses the order of products:
(AB)
T
=B
T
A
T
:
4
1.3. Applications of Matrices
In practice there is often a need for summing some of the entries of a matrix, in-
terchanging two rows (two columns), etc. Computers use matrix-multiplication
1.3.1. Summing Vectors
The entries of the rows of a matrix can be added by post-multiplying it by a
column-vector, whose all entries are 1's (by a summing vector). Similarly, by
pre-multiplying the matrix by a row-vector with all of its entries 1's, results in
the summing of the entries in the columns.
E.g.:
A =

1 4 3
2 5 3

; B =
2
4
1
1
1
3
5
; C =

1 1

AB =

2
0

; CA =

3 1 0

:
1.3.2. Permutation Matrices
As seen above (in multiplication property b), multiplying a matrix with the unit
matrix of suitable size results in no change of the original matrix. If the columns
of a unit matrix are interchanged (permutated), and a matrix is post-multiplied
by this permutation-matrix, then the suitable columns of the original matrix
will be permutated.
E.g.:

1 4 3
2 5 3


2
4
0 1 0
0 0 1
1 0 0
3
5
=

3 1 4
3 2 5

:
"
c
3
"
c
1
"
c
2
"
c
3
"
c
1
"
c
2
Similarly, if the rows of a unit matrix are interchanged (permutated), and a
matrix is pre-multiplied by this permutation-matrix, then the suitable rows of
the original matrix will be permutated.
E.g.:
r
2
!
r
1
!

0 1
1 0



1 4 3
2 5 3

=

2 5 3
1 4 3

r
2
r
1
5
```
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