Property 1 :
The value of a determinant is unaltered by interchanging its rows and columns.
Property 2 :
If any two rows or columns of a determinant are interchanged the determinant changes its sign but its numerical value is unaltered.
Corollary :
The sign of a determinant changes or does not change according as there is an odd or even number of interchanges among its rows or columns.
Property 3 :
If two rows or columns of a determinant are identical then the value of the determinant is zero.
Property 4 :
If every element in a row or column of a determinant is multiplied by a constant k then the value of the determinant is multiplied by k.
Deduction from properties (3) and (4) :
If two rows or columns of a determinant are proportional i.e. one row or column is a scalar multiple of other row or column then its value is zero.
Property 5 :
If every element in any row or column can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows or columns of both being the same.
Property 6 :
A determinant is unaltered when to each element of any row or column is added to those of several other rows or columns multiplied respectively by constant factors.
That is, a determinant is unaltered when to each element of any row or column is added by the equimultiples of any parallel row or column.
Without expanding, evaluate the following determinants.
Example 1 :
Solution :
Factor 3x from the third row.
In the above determinant, rows 1 and 3 are identical.
= 3(0)
= 0
So, the value of the given determinant is 0.
Example 2 :
Solution :
Add the first and second row and make the result as first row.
That is, R_{1} ----> R_{1} + R_{2}.
Factor (x + y + z) from the third row.
In the above determinant, rows 1 and 3 are identical.
= (x + y + z)(0)
= 0
So, the value of the given determinant is 0.
Example 3 :
If A is a square matrix and |A| = 2, find the value of |AA^{T}|.
Solution :
Determinant of A and determinant of A^{T} will be numerically equal.
|A| = |A^{T}|
We know that |AB| = |A||B|.
Then,
|AA^{T}| = |A||A^{T}|
= |A||A|
= |A|^{2}
= 2^{2}
= 4
Example 4 :
If A and B are square matrices of order 3 such that |A| = -1 and |B| = 3, find the value of |3AB|.
Solution :
For any square matrix, we have
|KA| = k^{n}|A|
(Here n stands for order of the matrix)
|3AB| = 3^{3}|AB|
Base 3 stands for k and power 3 stands for n that is order of the given matrix.
= 27|A||B|
= 27(-1) (3)
= -81
Example 5 :
If λ = -2, determine the value of
Solution :
Substitute λ = -2 in the above determinant.
The above is the determinant of a skew matrix.
For any skew matrix A, |A| = 0
So, the value of the above determinant is 0.
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