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Matrices

Lecture on Co-factor
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Cofactor of an element in a determinant

A = \( \begin{vmatrix}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{vmatrix} aij=Mij*(-1)^(i+j) \)

\(​cf(a32)=A32=M32*(-1)^(3+2) \)

\(-1* \begin{vmatrix}a11&a13\\a21&a23\end{vmatrix} \)
 

Sum of product of elements of rows with its corresponding cofactors = value of determinant

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Lecture on Minor
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For calculation of minor of an element aij  hide i th row and j th column|
Minor of aij = \(M11= \begin{vmatrix} a22& a23\\ a32 &a33\end{vmatrix}
\)
 
Let A = \( \begin{vmatrix}a&b&c&d\\ e&f&g&h\\i&j&k&l\\m&n&o&p\end{vmatrix}
minor(g)=\begin{vmatrix}a&b&d\\i&j&l\\m&n&p\end{vmatrix} \)

 

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Lecture on Multiplication
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Multiplication of Matrix with a scalar:

\(k\begin{vmatrix}a&b\\c&d\end{vmatrix} = \begin{vmatrix}ka&kb\\kc&kd\end{vmatrix} \)
Example:
\(2*\begin{vmatrix}7&3\\9&1\end{vmatrix} = \begin{vmatrix}14&6\\18&2 \end{vmatrix} \)

Multiplication of Matrix with with another matrix:
M=Aij X Bpq is possible only when j = p and order of M is Mjk

Properties of Matrix Multiplication:

  1. k(A + B) = kA +kB
  2. p(qA) = (pq)A = q(pA)
  3. -p(A) = -(pA) = p(-A)
  • Important points about matrix multiplication:
  1. Not commutative
    AB ≠ BA
  2. Associative
    A(BC) = (AB)C
  3. Distributive over addition 
    A(B+C) = AB + AC
  4. AB = 0  doesn't mean  A = 0 or B = 0
  5. AB = AC ⇒ B = C iff A is non-singular matrix.
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Lecture on Trace
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Trace of a matrix is the sum of its principal diagonal elements.

Example:\( \begin{bmatrix}2&7&9\\3&4&6\\5&6&-5\end{bmatrix} \) trace = 2 + 4 + -5 =1

Necessary condition for trace calculation is matrix should be a square matrix.

Properties of trace are:

  1. tr(λA) = λ*tr(A)
  2. tr(A+B) = tr(A) +tr(B)
  3. tr(AB) = tr(BA)
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Lecture on Transpose
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Transpose of a matrix is a matrix formed by interchanging rows and columns.

\( A= \begin{bmatrix}a&b&c\\d&e&f\end{bmatrix} \) then \( A'= \begin{bmatrix}a&d\\b&e\\c&f\end{bmatrix} \)

Properties of transpose are:

  1. \((A')'=A\)
  2. \((A+B)'=A'+B'
    \)
  3. \(k(A')=(kA)'
    \)
  4. \((AB)'=B'A'\)
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Lecture on Conjugate of Matrix
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Conjugate of Matrix (\(A^-\)):

\(A= \begin{bmatrix}3+2i&6+i&4\\4-i&7+2i&3-4i\end{bmatrix} \) then , \(A^-= \begin{bmatrix}3-2i&6-i&4\\4+i&7-2i&3+

4i\end{bmatrix} \) 

Properties of conjugate of matrix are:

  1. \(A^-=A
    \)
  2. \((A+B)^-=A^-+B^-\)
  3. \((KA)^-=K^-A^-
    \)
  4. \((AB)^-=A^-B^-\)
  5. If \((A^-)=-A\) // It means matrix contains only imaginary values

Transpose conjugate (\(A^ \Theta
\)
) = (\(A^-\))'

Properties of transpose conjugate:

  1. \((A^ \Theta )^ \Theta
    \)
    = A
  2. (\((A+B )^ \Theta
    \)
    ) = \((A^ \Theta ) +(B
    ^ \Theta )
    \)
  3. \((AB )^ \Theta=B^ \Theta A^ \Theta\)
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