FAQs on Examples : Based on Properties of Determinants Video Lecture  Mathematics (Maths) Class 12  JEE
1. What are the properties of determinants? 

Ans. The properties of determinants include:
 If two rows (or columns) of a determinant are interchanged, the value of the determinant changes its sign.
 If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
 If all the elements of any row (or column) of a determinant are multiplied by a scalar k, then the value of the determinant is also multiplied by k.
 If each element of a row (or column) of a determinant is expressed as the sum of two terms, then the determinant can be expressed as the sum of two determinants.
2. How can determinants be used to solve systems of linear equations? 

Ans. Determinants can be used to solve systems of linear equations by using Cramer's Rule. Cramer's Rule states that if we have a system of linear equations in the form Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector, then the solution can be found by taking the determinants of various matrices.
To solve for the variables, we calculate the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constant vector b. Then, the solution for each variable is given by the ratio of the determinant of the corresponding matrix to the determinant of the coefficient matrix.
3. Can determinants be used to find the area of a triangle? 

Ans. Yes, determinants can be used to find the area of a triangle. Given the coordinates of the three vertices of a triangle, we can form a matrix using these coordinates. By taking half the absolute value of the determinant of this matrix, we can find the area of the triangle.
The formula to find the area of a triangle using determinants is:
Area = 0.5 * (x1 * (y2  y3) + x2 * (y3  y1) + x3 * (y1  y2))
Here, (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle.
4. Can determinants be negative? 

Ans. Yes, determinants can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix. If the number of row or column interchanges required to transform the matrix into its rowechelon form is odd, then the determinant will be negative. If the number of row or column interchanges is even, the determinant will be positive.
For example, if we have a 2x2 matrix [a b; c d], the determinant is given by ad  bc. If ad  bc is negative, then the determinant itself is negative.
5. Are determinants only used in linear algebra? 

Ans. No, determinants are not only used in linear algebra. While determinants have a significant role in linear algebra, they also find applications in various other fields, such as calculus, physics, statistics, and computer science.
In calculus, determinants are used to find the Jacobian matrix, which is used in multivariable calculus and change of variables.
In physics, determinants are used to calculate the moment of inertia of objects and to solve problems involving rotational motion.
In statistics, determinants are used in multivariate analysis, such as in the calculation of covariance matrices.
In computer science, determinants are used in graphics and image processing algorithms, such as transformations and image compression.