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exercise - 4.1 🔥 | determinants 👌🏼| #determinants of #matrices of order n x n | NCERT #class12mathsinhindi




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3. If A = [■8(𝟏&𝟐@𝟒&𝟐)], then show that |2A| = 4 |A|.




4. If A = [■8(𝟏&𝟎&𝟏@𝟎&𝟏&𝟐@𝟎&𝟎&𝟒)], then show that |3 A| = 27 |A|.




5. Evaluate the determinants:
(i) |■8(𝟑&−𝟏&−𝟐@𝟎&𝟎&−𝟏@𝟑&−𝟓&𝟎)| (ii) |■8(𝟑&−𝟒&𝟓@𝟏&𝟏&−𝟐@𝟐&𝟑&𝟏)|




(iii) |■8(𝟎&𝟏&𝟐@−𝟏&𝟎&−𝟑@−𝟐&𝟑&𝟎)| (iv) |■8(𝟐&−𝟏&−𝟐@𝟎&𝟐&−𝟏@𝟑&−𝟓&𝟎)|




6. If A = [■8(𝟏&𝟏&−𝟐@𝟐&𝟏&−𝟑@𝟓&𝟒&−𝟗)], find |A|.



7. Find values of x, if
(i) |■8(𝟐&𝟒@𝟓&𝟏)|= |■8(𝟐𝒙&𝟒@𝟔&𝒙)| (ii) |■8(𝟐&𝟑@𝟒&𝟓)|= |■8(𝒙&𝟑@𝟐𝒙&𝟓)|




8. If |■8(𝒙&𝟐@𝟏𝟖&𝒙)|= |■8(𝟔&𝟐@𝟏𝟖&𝟔)|, then x is equal to
(A) 6 (B) ± 6 (C)– 6 (D) 0






DETERMINANTS:

In class 12 mathematics, determinants are a fundamental concept associated with square matrices. They represent a scalar value that can be calculated from the elements of a square matrix. Determinants are useful for solving systems of linear equations and have applications in various fields like physics and engineering.

Here's a more detailed explanation:

What is a determinant?
A determinant is a scalar value derived from the elements of a square matrix.
It's denoted by "det(A)" or "|A|", where A is the matrix.

For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated as ad - bc.

For a 3x3 matrix, the determinant is calculated using a specific formula involving the elements of the matrix.

Only square matrices (matrices with the same number of rows and columns) have determinants.

Key Concepts related to Determinants in Class 12:

Minors:
The minor of an element in a determinant is the determinant formed by deleting the row and column containing that element.

Cofactors:
The cofactor of an element is its minor multiplied by a sign (+ or -) determined by its position in the matrix.

Properties of Determinants:
There are several properties of determinants that can be used to simplify calculations, such as the property of interchanging rows or columns, multiplying a row or column by a constant, and adding a multiple of one row to another.

Applications:
Determinants are used to:

Find the area of a triangle.
Solve systems of linear equations (e.g., using Cramer's rule).
Determine if a matrix is invertible (non-singular).
Find the inverse of a matrix.
Calculate the adjoint of a matrix.
In essence, determinants are a powerful tool in linear algebra that provide valuable information about square matrices and their associated systems of equations.