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| === Interchange Property === | | === Interchange Property === |
| The value of a determinant remains unchanged if the rows or the columns of a determinant are interchanged. | | The value of a determinant remains unchanged if the rows and the columns of a determinant are interchanged. |
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| <math>\bigtriangleup= \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math> <math>\bigtriangleup_1= \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\a_3 & b_3 & c_3 \end{vmatrix}</math> | | Before the rows and the columns are interchanged |
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| | <math>\bigtriangleup= \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math> |
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| | After the rows and the columns are interchanged |
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| | <math>\bigtriangleup_1= \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\a_3 & b_3 & c_3 \end{vmatrix}</math> |
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| <math>\bigtriangleup=\bigtriangleup_1</math> | | <math>\bigtriangleup=\bigtriangleup_1</math> |
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| <math>\bigtriangleup= a_1b_2c_3-a_1b_3c_2 - a_2b_1c_3+a_2b_3c_1+a_3 b_1c_2-a_3b_2c_1 </math> | | <math>\bigtriangleup= a_1b_2c_3-a_1b_3c_2 - a_2b_1c_3+a_2b_3c_1+a_3 b_1c_2-a_3b_2c_1 </math> |
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| === Sign Property === | | === Sign Property === |
| | If any two rows or any two columns are interchanged, the sign of the value of the determinant changes. |
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| | <math>\bigtriangleup= \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math> |
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| | After changing any two rows |
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| | <math>\bigtriangleup_1= \begin{vmatrix} a_1 & a_2 & a_3 \\c_1 & c_2 & c_3 \\ b_1 & b_2 & b_3 \end{vmatrix}</math> |
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| | <math>\bigtriangleup=\bigtriangleup_1</math> |
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| | '''Verification''' |
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| | <math>\bigtriangleup= a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \end{vmatrix}</math> |
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| | <math>\bigtriangleup= a_1 (b_2c_3-b_3c_2) - a_2 (b_1c_3-b_3c_1)+a_3 (b_1c_2-b_2c_1) </math> |
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| | <math>\bigtriangleup= a_1b_2c_3-a_1b_3c_2 - a_2b_1c_3+a_2b_3c_1+a_3 b_1c_2-a_3b_2c_1 </math> |
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| | <math>\bigtriangleup= a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \end{vmatrix}</math> |
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| === Zero Property === | | === Zero Property === |
Revision as of 09:56, 24 January 2024
Properties of determinants are required to find the value of the determinant with least calculations. The properties of determinants are based on the elements, the row, and column operations, and it helps to easily find the value of the determinant.
Properties of Determinants
Interchange Property
The value of a determinant remains unchanged if the rows and the columns of a determinant are interchanged.
Before the rows and the columns are interchanged
After the rows and the columns are interchanged
Verification
Hence
If the rows and columns of the matrix are interchanged, then the transpose of the matrix is obtained and the determinant value and the determinant of the transpose are equal.
Sign Property
If any two rows or any two columns are interchanged, the sign of the value of the determinant changes.
After changing any two rows
Verification
Zero Property
Multiplication Property
Sum Property
Property of Invariance
Triangular Property