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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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| <math>\bigtriangleup_1= a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1 \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1 \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}</math> | | <math>\bigtriangleup_1= a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1 \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1 \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}</math> |
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| === Sum Property === | | === Sum Property === |
| | If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants. |
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| | <math>\begin{vmatrix} a_1+d_1 & a_2+d_2 & a_3+d_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}=\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix} + \begin{vmatrix} d_1 & d_2 & d_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math> |
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| | '''Verification''' |
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| | L.H.S =<math>\begin{vmatrix} a_1+d_1 & a_2+d_2 & a_3+d_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math> |
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| | <math>=(a_1+d_1)(b_2c_3-b_3c_2)- (a_2+d_2) (b_1c_3-b_3c_1)+(a_3+d_3)(b_1c_2-b_2c_1) </math> |
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| | <math>=a_1(b_2c_3-b_3c_2)-a_2 (b_1c_3-b_3c_1)+a_3(b_1c_2-b_2c_1) + |
| | d_1(b_2c_3-b_3c_2)-d_2 (b_1c_3-b_3c_1)+d_3(b_1c_2-b_2c_1) </math> |
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| | <math>=\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix} + \begin{vmatrix} d_1 & d_2 & d_3 \\ b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}</math>=R.H.S |
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| === Property of Invariance === | | === Property of Invariance === |
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| === Triangular Property === | | === Triangular Property === |
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
If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.
Verification
Multiplication Property
If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k
Verification
Sum Property
If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
Verification
L.H.S =
=R.H.S
Property of Invariance
Triangular Property