Properties of determinants

Alll the properties of determinants

Consider the equations a₁x+b₁y = 0 and  a₂x+b₂y = 0. These give 

⇒ ⇒ a₁b₂ – a₂b₁ = 0

We express this eliminate as = 0.

A determinant of order three consisting of 3 rows and 3 columns is written as  and is equal to

The numbers ai, bi, ci  ( i =1,2,3 ) are called the elements of the determinant. Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah. 

properties of determinants -1 

If rows be changed into columns and columns into the rows, the determinant remains unaltered.

properties of determinants -2 

If any two row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.

Note-If any line of a determinant D be passed over ‘m’ parallel lines, the resulting  determinant D′ is equal to (-1)m D

 D′ =   ⇒  D′ =    

⇒    D′ = (-1) 2 D =D.

properties of determinants -3  

If two rows (or two columns) in a determinant have corresponding entries that are equal (or proportional), the value of determinant is equal to zero.

properties of determinants –4

If each of the entries of one row (or columns) of a determinant is multiplied by a nonzero constant k, then the determinant gets multiplied by k.

properties of determinants -5

If each entry in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.

properties of determinants -6

If to each element of a line (row or column) of a determinant be added the equimutiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered   =

properties of determinants –7

If each entry in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.

properties of determinants -8

If a determinant D vanishes for x = a, then (x-a) is a factor of D, In other words, if two     rows (or two columns) become identical for x = a. then (x- a) is a factor of D.

For example, let D = , if we are putting a = b, we will get D = 0. i.e. a – b is a factor of D.

Note:   

In general, if r rows (or r columns) become identical when a is substituted for x, then
(x-a)^r-1 is a factor of D.   

properties of determinants -9

If in a determinant (of order three or more) the elements in all the rows (columns) are in A.P. with same or different common difference, the value of the determinant is zero.

Additional Note

• It is important to know that all the properties applicable to rows are also equally applicable to columns but independently
• Whenever rows are disturbed by applications of properties of determinants, at least one of the row shall remain in original shape. In other words all the rows shall not be disturbed at a time.
• It is always desirable to try to bring in as many zeros as possible in any row ( or column) and then expand the determinant with respect to that row (column). Mere expansion from the outset should be avoided as far as possible.

 We can express a determinant as

Where Ci ( i = 1,2, 3 ) are the columns and Rj ( j=1,2,3) are the rows of the determinant.

Example: Show that Δ = 0  if Δ = .

Detail Explanation :    Operating C₂ → C₂ – ( C1 + C₃) , we get

    Δ = = 0

Note: Using  the A.P. property  one can immediately write Δ = 0 directly