write the further use of determinant
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Pre-requisite :-
A determinant can be defined in many ways for a square matrix.
The first and most simple way is to formulate the determinant by taking into account the top row elements and the corresponding minors. Take the first element of the top row and multiply it by it’s minor, then subtract the product of the second element and its minor. Continue to alternately add and subtract the product of each element of the top row with its respective minor until all the elements of the top row have been considered.
Properties of Determinant :-
USES :-
There are 10 main properties of determinants which include reflection property, all-zero property, proportionality or repetition property, switching property, scalar multiple property, sum property, invariance property, factor property, triangle property, and co-factor matrix property.
1. Reflection Property:
The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.
2. All-zero Property:
If all the elements of a row (or column) are zero, then the determinant is zero.
3. Proportionality (Repetition) Property:
If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.
4. Switching Property:
The interchange of any two rows (or columns) of the determinant changes its sign.
5. Scalar Multiple Property:
If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
6. Sum Property:
[tex]\left[\begin{array}{ccc}a1+b1&c1&d1\\a2+b2&c2&d2\\a3+b3&c3&d3\end{array}\right] =\left[\begin{array}{ccc}a1&c1&d1\\a2&c2&d2\\a3&c3&d3\end{array}\right] +\left[\begin{array}{ccc}b1&c1&d1\\b2&c2&d2\\b3&c3&d3\end{array}\right][/tex]
7. Property of Invariance:
[tex]\left[\begin{array}{ccc}a1&b1&c1\\a2&b2&c2\\a3&b3&c3\end{array}\right] =\left[\begin{array}{ccc}a1+\alpha b1+\beta c1 &b1&c1\\a2+\alpha b2+\beta c2&b2&c2\\a3+\alpha b3+\beta c3&b3&c3\end{array}\right][/tex]
That is, a determinant remains unaltered under an operation of the form Ci→Ci + αCj + βCk, where j,k ≠ i, or an operation of the form
Ri→Ri + αRj + βRk, where j,k≠i.
8. Factor Property:
If a determinant Δ becomes zero when we put x=α, then (x−α) is a factor of Δ.
9. Triangle Property:
If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements. That is,
[tex]\left[\begin{array}{ccc}a1&a2&a3\\0&b2&b3\\0&0&c3\end{array}\right] =\left[\begin{array}{ccc}a1&0&0\\a2&b2&0\\a3&b3&c3\end{array}\right] =a1b2c3[/tex]