# Determinants Study Notes

By Akhil Gupta|Updated : November 7th, 2020

### Determinant

1.1.   Definition:

The expression

is called a determinant of the second order and stands for ‘a11b22 – a21b12’. It contains 4 numbers a11, b12, a22, b22 (called elements) which are arranged along two horizontal lines (called rows) and two vertical lines (called columns).

Similarly,

is called a determinant of the third order. It consists of 9 elements which are arranged in 3 rows and 3 columns.

1.1.1. Principal Diagonal:

The diagonal through the left-hand top corner which contains the elements a11, b22, c33, …... is called the leading or principal diagonal.

1.2.   Minor and Cofactors:

1.2.1. Minor:

The minor of the element aij is denoted Mij and is the determinant of the matrix that remains after deleting row i and column j of A.

1.2.2. Co – factor:

The cofactor of aij is denoted Cij and is given by:

Cij = (–1) i+j Mij

The cofactor of an element is usually denoted by the corresponding capital letter.

1.3 Properties of Determinant:

(a). The value of a determinant does not change when rows and columns are interchanged i.e.

|AT| = |A|

(b). If any row (or column) of a matrix A is completely zero, then:

|A| = 0, Such a row (or column) is called a zero row (or column).

(c). Also, if any two rows (or columns) of a matrix A are identical, then |A| = 0.

(d). If any two rows or two columns of a determinant are interchanged the value of determinant is multiplied by –1.

(e). If all elements of the one row (or one column) or a determinant are multiplied by same number k the value of determinant is k times the value of given determinant.

(f). If A be n-rowed square matrix, and k be any scalar, then |kA| = kn|A|.

(g). (i) In a determinant the sum of the products of the element of any row (or column) with the cofactors of corresponding elements of any row or column is equal to the determinant         value.

(ii) In determinant the sum of the products of the elements of any row (or column) with the cofactors of some other row or column is zero.

(h). If to the elements of a row (or column) of a determinant are added k times the corresponding elements of another row (or column) the value of determinant thus obtained is equal to the value of original determinant.

(i). |AB| = |A|×|B| and based on this we can prove the following: |An| = (|A|)n

(j). Using the fact that A · Adj A = |A|. I, the following can be proved for An×n.

Thanks

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