NCERT Class 12-Mathematics: Exemplar Chapter –4 Determinants Part 1

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4.1 Overview

To every square matrix of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where is the element of A.

If A , then determinant of A, denoted by (or det A), is given by

Remarks

(i) Only square matrices have determinants.

(ii) For a matrix A, A is read as determinant of A and not, as modulus of A.

4.1.1 Determinant of a matrix of order one

Let be the matrix of order 1, then determinant of A is defined to be equal to .

4.1.2 Determinant of a matrix of order two

Let be a matrix of order . Then the determinant of A is defined as: det .

4.1.3 Determinant of a matrix of order three

The determinant of a matrix of order three can be determined by expressing it in terms of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order corresponding to each of three rows and three columns and each way gives the same value.

Consider the determinant of a square matrix , i.e.,

Expanding along , we get

Remark In general, if , where A and B are square matrices of order , then .

4.1.4 Properties of Determinants

For any square matrix satisfies the following properties.

(i) , where transpose of matrix .

(ii) If we interchange any two rows (or columns), then sign of the determinant changes.

(iii) If any two rows or any two columns in a determinant are identical (or proportional), then the value of the determinant is zero.

(iv) Multiplying a determinant by k means multiplying the elements of only one row (or one column) by k.

(v) If we multiply each element of a row (or a column) of a determinant by constant , then value of the determinant is multiplied by .

(vi) If elements of a row (or a column) in a determinant can be expressed as the sum of two or more elements, then the given determinant can be expressed as the sum of two or more determinants.

(vii) If to each element of a row (or a column) of a determinant the equimultiples of corresponding elements of other rows (columns) are added, then value of determinant remains same.

Notes:

(i) If all the elements of a row (or column) are zeros, then the value of the determinant is zero.

(ii) If value of determinant ‘’ becomes zero by substituting , then is a factor of ‘’.

(iii) If all the elements of a determinant above or below the main diagonal consists of zeros, then the value of the determinant is equal to the product of diagonal elements.

4.1.5 Area of a triangle

Area of a triangle with vertices and is given by

4.1.6 Minors and co-factors

(i) Minor of an element of the determinant of matrix A is the determinant obtained by deleting row and column, and it is denoted by .

(ii) Co-factor of an element is given by .

(iii) Value of determinant of a matrix A is obtained by the sum of products of elements of a row (or a column) with corresponding co-factors. For example

(iv) If elements of a row (or column) are multiplied with co-factors of elements of any other row (or column), then their sum is zero. For example,

.

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