DETERMINANTS

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Scholars learning

• DETERMINANTS

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• Every square matrix can be associated to an
  expression or a number which is known as its
  determinant.
•  
• If A is a square matrix of order 2 X 2,
  then its determinant is denoted by
• A or, and is defined as a11 a22 a12 a21.
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• i.e. A a11 a22 a12 a21
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• If A is a square matrix of order 3 X
  3,
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• then its determinant is denoted by A or,
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• and is equal to a11 a22 a33 a12 a23 a31 a13
  a32 a21 a11 a23 a32 - a22 a13 a31 a12
  a21 a33

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• This expression can be arranged in the following
  form
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• (-1)1 1 a11 (-1)1 2 a12
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• (-1)1 3 a13
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• This is known as the expansion of A along first
  row.
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• In fact, A can be expanded along any of its
  rows or columns. In order to expand A along any
  row or column, we multiply
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• Example 1 - Evaluate the determinant
• D by expanding it along first column.
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• SOLUTION By using the definition, of expansion
  along first column, we obtain
• D
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• D (-1)11 (2) (-1)21 (1)
  (-1)31 (-2)
• D 2 - -2
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• D 2 (-6-3) (-92) -2(94) -18 7-26 -37.
• NOTE 1 Only square matrices have their
  determinants. The matrices which are not square
  do not have determinants.
• NOTE 2 The determinant of a square matrix of
  order 3 can be expressed along any row or column.
• NOTE 3 If a row or a column of a determinant
  consists of all zeros, then the value of the
  determinant is zero.
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• There are three rows and three columns in a
  square matrix of order 3.
• PROPERTIES OF DETERMINANTS
• We have defined the determinants of a square
  matrix of order 4 or less. In fact, these
  definitions are consequences of the general
  definition of the determinant of a square matrix
  of any order which needs so many advanced
  concepts. These concepts are beyond the scope of
  this book. Using the said definition and some
  other advanced concepts we can prove the
  following properties. But, the concepts used in
  the definition itself are very advanced.
  Therefore we mention and verify them for a
  determinant of a square matrix of order 3.

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• Property 1 let A aij be a square matrix of
  order n, then the sum of the product of elements
  of any row(column) with their cofactors is always
  equal to A or, det (A).
• Property 2 let A aij be a square matrix of
  order n, then the sum of the product of elements
  of any row(column) with the cofactors of the
  corresponding elements of some other row (column)
  is zero.
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• Property 3 Let A aij be a square matrix of
  order n, then A AT.
• Property 4 let A aij be a square matrix of
  order n(2) and let B be a matrix obtained from A
  by interchanging any two rows(columns) of A, then
  B -A.
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• Conventionally this property is also stated as
• If any two rows (columns) of a determinant are
  interchanged, then the value of the determinant
  changes by minus sign only.
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• Property 5 if any two rows (columns) of a square
  matrix A aij of order n (gt2) are identical,
  then its determinant is zero i.e. A 0.
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• Property 6 Let A aij be a square matrix of
  order n, and let B be the matrix obtained from A
  by multiplying each element of a row (column) of
  A by a scalar k, then B k A.

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• Property 7 Let A square matrix such that each
  element of row (column) of A is expressed as the
  sum of two or more terms. Then, the determinant
  of A can be expressed as the sum of the
  determinants of two or more matrices of the same
  order.