# Determinants Class 12 Notes- Chapter 4

## What is Determinant of a matrix?

When a square matrix “A” of an order “n” is associated with a number, then it is titled as a determinant of the aforementioned matrix. The number involved in this square matrix can be a real number or a complex number.

In the field of mathematics, Determinants can be used for a myriad of different calculations such as:

• Finding the area of a Triangle.
• Obtaining the solutions of linear equations in two or three variables.
• Solve Linear equations by using the inverse of a matrix
• Getting the adjoint and the inverse of a square matrix

### What are the properties of Determinants?

The properties of determinants have its uses, in simplifying the evaluation by getting the maximum number of zeroes in a row or column. They hold true for determinants of any order. The properties of determinants are mentioned below:

1. The determinant’s value remains unchanged if the rows and columns are interchanged.
2. The sign of a determinant changes when any two rows or columns of a determinant are interchanged with each other.
3. The value of a determinant is zero when any of the rows or columns of a determinant are identical to each other.
4. For each element of a row or column which gets multiplied by a constant P, the value also gets multiplied by P.
5. The determinant obtained is a sum of two or more determinants if some or all elements of a column or a row are expressed as a sum of two or more terms
6. When the equimultiples of corresponding elements are added, the value of the determinant remains the same, whenever the operation is multiplied.

### Important Questions:

1. Examine the consistency of the system of equations
1. x + 2y = 2, 2x + 3y = 3
2. 2x – y = 5, x + y = 4
3. x + 3y = 5, 2x + 6y = 8
2. Using Cofactors of elements of second row, evaluate ∆= $\begin{vmatrix} 5 &3 &8 \\ 2 &0 &1 \\ 1 &2 &3 \end{vmatrix}$
3. Find values of p if area of triangle is 6 sq. units and vertices are
(i) (p, 0), (6, 0), (0, 4)
(ii) (–4, 0), (0, 6), (0, p)
4. Using determinants, find equation of line joining (2, 3) and (4, 7)
5. Which of the following is correct
(A) Determinant is a number associated to a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a square matrix.
(D) None of these