# Properties Of Determinants

### 1. First Property

The value of the determinant remains unchanged if its rows and columns are interchanged.

eg A Matrix A is given by, A=

Determinant of above matrix is D1= a11 (a22*a33 - a23*a32)  -a12(a21*a33 - a23*a31) + a13(a21*a32 - a22*a31)

Lets make a matrix where columns and rows are interchanged, Then above matrix becomes:

Determinant of above is  equal to D2= a11 (a22*a33 - a23*a32)  -a12(a21*a33 - a23*a31) + a13(a21*a32 - a22*a31)

Hence D1=D2

### 2. Second Property

If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

eg Lets take the matrix A from above=

Determinant of A is given by D1= a11 (a22*a33 - a23*a32)  -a12(a21*a33 - a23*a31) + a13(a21*a32 - a22*a31)

Lets change the rows 2 and 3 of above matrix, Then Matrix A becomes

Determinant of above matrix is now D2= a21(a32*a13) - a22(a31*a13 - a33*a11) + a23(a31*a12 - a32*a11)

D2 can also be written as D2= -a11 (a22*a33 - a23*a32)  + a12(a21*a33 - a23*a31) - a13(a21*a32 - a22*a31)

Thus D1 = -D2

### 3. Third Property

If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

eg consider the two matrix A,

A =

Determinatn of A , DETa= 1 ( 5*6 - 5*6) - 2(4*6 - 6*4) + 3(4*5 - 5*4)

DET = 0

### 4. Fourth Property

If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

eg COnsider the follwong Matrix A,

A=

Determinant D = a1(b2*c3 - c2*b3) - b1( a2*c3 - c2*a3) + c1( a2*b3 - b2*a3)

if Row 2 is multiplied by scaler value k, hence a Matrix B is formed,

B=

Determinant of B is equalt to, D= a1(kb2*c3 - kc2*b3) - b1( ka2*c3 - kc2*a3) + c1(k a2*b3 - kb2*a3)

hence D = k *( a1(b2*c3 - c2*b3) - b1( a2*c3 - c2*a3) + c1( a2*b3 - b2*a3) )

Determinant of B = k * Determinant of A