Difference between revisions of "Determinant"
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== Determinant of a matrix of order one == | == Determinant of a matrix of order one == | ||
Let <math>A = \begin{bmatrix} a \end{bmatrix}</math> be the matrix of order <math>1</math>, then determinant of <math>A</math> is defined to be equal to <math>a</math> | Let <math>A = \begin{bmatrix} a \end{bmatrix}</math> be the matrix of order <math>1</math>, then determinant of <math>A</math> is defined to be equal to <math>a</math> | ||
=== Example === | |||
<math>\begin{vmatrix}A \end{vmatrix}=\begin{vmatrix}2 \end{vmatrix}=2</math> | |||
== Determinant of a matrix of order two == | == Determinant of a matrix of order two == | ||
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<math>1 \times a_{13}\begin{vmatrix}a_{21} & a_{22}\\ a_{31} & a_{32} \end{vmatrix} </math> | <math>1 \times a_{13}\begin{vmatrix}a_{21} & a_{22}\\ a_{31} & a_{32} \end{vmatrix} </math> | ||
'''Step 4:''' The expansion of determinant of <math>A</math>( <math>\begin{vmatrix}A \end{vmatrix}</math>)written as sum of all three terms obtained in steps 1, 2 and 3 above is given by | '''Step 4:''' The expansion of determinant of <math>A</math> ( <math>\begin{vmatrix}A \end{vmatrix}</math>) written as sum of all the three terms obtained in steps 1, 2 and 3 above is given by | ||
<math>\begin{vmatrix}A \end{vmatrix}=(1 \times a_{11}\begin{vmatrix}a_{22} & a_{23}\\ a_{32} & a_{33} \end{vmatrix})+(-1 \times a_{12}\begin{vmatrix}a_{21} & a_{23}\\ a_{31} & a_{33} \end{vmatrix})+(1 \times a_{13}\begin{vmatrix}a_{21} & a_{22}\\ a_{31} & a_{32} \end{vmatrix}) </math> | <math>\begin{vmatrix}A \end{vmatrix}=(1 \times a_{11}\begin{vmatrix}a_{22} & a_{23}\\ a_{32} & a_{33} \end{vmatrix})+(-1 \times a_{12}\begin{vmatrix}a_{21} & a_{23}\\ a_{31} & a_{33} \end{vmatrix})+(1 \times a_{13}\begin{vmatrix}a_{21} & a_{22}\\ a_{31} & a_{32} \end{vmatrix}) </math> | ||
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<math>\begin{vmatrix}A \end{vmatrix}=(1 \times 3\begin{vmatrix}-2 & 5\\ 8 & 7 \end{vmatrix})+(-1 \times 1\begin{vmatrix}4 & 5\\ 2 & 7 \end{vmatrix})+(1 \times 1\begin{vmatrix}4 & -2\\ 2 & 8 \end{vmatrix}) </math> | <math>\begin{vmatrix}A \end{vmatrix}=(1 \times 3\begin{vmatrix}-2 & 5\\ 8 & 7 \end{vmatrix})+(-1 \times 1\begin{vmatrix}4 & 5\\ 2 & 7 \end{vmatrix})+(1 \times 1\begin{vmatrix}4 & -2\\ 2 & 8 \end{vmatrix}) </math> | ||
<math>\begin{vmatrix}A \end{vmatrix}= 3(-2 \times 7 -5 \times 8)-1(4 \times 7-5 \times 2)+1(4 \times 8-(-2 \times 2) </math> | <math>\begin{vmatrix}A \end{vmatrix}= 3(-2 \times 7 -5 \times 8)-1(4 \times 7-5 \times 2)+1(4 \times 8-(-2 \times 2)) </math> | ||
<math>\begin{vmatrix}A \end{vmatrix}= 3(-14 -40)-1(28-10)+1(32-(-4)) </math> | <math>\begin{vmatrix}A \end{vmatrix}= 3(-14 -40)-1(28-10)+1(32-(-4)) </math> |
Latest revision as of 08:14, 16 January 2024
A determinant is a scalar value that can be calculated from the elements of a square matrix.
Definition
For every square matrix, of order a determinant can be defined as a scalar value that is real or a complex number, where
is the th element of matrix . The determinant can be denoted as or .
Determinant is written by taking the grid of numbers and arranging them inside the absolute-value bars instead of using square brackets.
Consider a matrix
Then, its determinant can be shown as:
Determinant of a matrix of order one
Let be the matrix of order , then determinant of is defined to be equal to
Example
Determinant of a matrix of order two
Let be the matrix of order , then determinant of is defined as
Example
Determinant of a matrix of order three
Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column).
Expansion of a determinant along a row ( First row)
Step 1: Multiply first element of first row by and with the second order determinant obtained by deleting the elements of first row and first column of as lies in first row and first column.
Step 2: Multiply first element of first row by and with the second order determinant obtained by deleting the elements of first row and second column of as lies in first row and second column.
Step 3: Multiply first element of first row by and with the second order determinant obtained by deleting the elements of first row and third column of as lies in first row and third column.
Step 4: The expansion of determinant of ( ) written as sum of all the three terms obtained in steps 1, 2 and 3 above is given by
Example