mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Module 3 Matrices
Lecture : 15
Rank of a matrix and linear dependence
The maximum number of linearly independent row vectors of a matrix 𝐴 = [𝑎𝑗𝑘 ] is
3 0 2 2
called the rank of A. e.g. [𝐴] = [−6 42 24 54] has rank 2.
21 − 21 0 15
This is because there exists a relation involving all three, 6𝑎⃗1 − 2
𝑎⃗2 − 𝑎⃗3
= 0
Where 𝑎⃗𝑖 is a vector constituting the row elements. For example,
1
𝑎⃗1 = ( ) , 𝑎⃗2 = ( ) , 𝑎⃗3 = ( )
3 −6 21
0 42 −21
2
2
24
54
0
15
It is to be kept in mind that the rank of a matrix is only zero when [𝐴] = 0. In another
example,
2 2 − 1
Consider a matrix [ 4 0 2 ]
0 6 3
where there is no such relation among the rows (or columns) and hence has a rank is 3.
Inverse of a Matrix
The inverse of a matrix can be defined in the same spirit as that of the inverse
of a
number. For example, the inverse of ‘2’ is 1
, which is also called as the reciprocal.
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2
However such divisions by matrices are not allowed. Still in the same way as a
number
and its reciprocal, when multiplied yields 1, a matrix and its inverse when multiplied
will yield a unit matrix 1. Thus inverse of a matrix A is defined as,
𝐴 × 𝐴−1 = 1
Where 𝐴−1 is the inverse of A and vice versa. The method of calculating
inverse of a matrix is shown below.

2. NPTEL – Physics – Mathematical Physics - 1
[𝐴−1] = 1
det[𝐴]
[𝐴 ]𝑇 =
𝑗
𝑘
1
det[𝐴] .
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[𝐴1𝑛 𝐴2𝑛 … … … … … 𝐴𝑛𝑛]
matrix A. Taking the transpose of a matrix is
equivalent to interchanging rows and columns. It is clear
that a matrix is invertible if the determinant exists. Here 𝐴𝑖𝑗
are the cofactors of the matrix A. Cofactors are defined
by the determinants of the matrix composed of matrix
elements that exclude the row and the column of
matrix element whose cofactor is being calculated.
1 2 3
For example, the cofactors of the matrix 𝐴 = [0 4 5]
1 0 6
𝐴12𝐴22 … … … … … 𝐴𝑛2
𝐴11𝐴21 … … …… … 𝐴𝑛1
.
.
where 𝐴𝑇 is the transpose of the
𝐴11 = |4 5
0 6
| = 24 𝐴31 = | | = −2
2 3
4 5
𝐴12 = − | | = 5
0 5
1 6
𝐴32 = − | | = −5
1 3
0 5
𝐴13 = |0 4
1 0
| = −4 𝐴33 = | | = 1
1 2
0 4
𝐴21 = − | | = −12
2 3
0 6
Thus the cofactor matrix is
24 5 − 4
|−12 3
2|
−2 − 5 4
𝐴22 = | | = 3
1 3
1 6
𝐴23 = − | | = 2
1 2
1 0
Using this, find the inverse of
−1 1 2
[𝐴] = [3 − 1 1], det [𝐴] = 10
−1 3 4

3. NPTEL – Physics – Mathematical Physics - 1
More examples :
1 1 1
𝐴 = ( 1 2 1)
1 1 3
1) First calculate det A to ascertain A is invertible.
2) The cofactors of the top row are
𝑀11 = | | = 5, 𝑀12 = | | = 2, 𝑀13 = | | = −1
2 1 1 1 1 2
1 3 1 3 1 1
Thus, det 𝐴 = 𝐴11𝑀11 − 𝐴12𝑀12 + 𝐴13𝑀13 = 2
= 2, thus A is invertible. (Since ≠ 0)
3) The cofactor matrix is constructed from the
minor,
𝐶 = (−𝑀21𝑀22 − 𝑀23) = ( −2
𝑀11 − 𝑀12𝑀13
𝑀31 − 𝑀32𝑀33
5 − 2 − 1
2 0 )
−1 0 1
Thus, 𝐴−1 = 1
( −2
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2
5 − 2 − 1
0 )
1
2
0
−1