With every square matrix, we can associate a number which is called determinant of matrix. It is denoted by |A| for matrix A. In this post, we will learn how to calculate determinant of 1 x 1, 2 x 2 and 3 x3 matrices. We can also calculate value of determinant of different square matrices with the help of co-factors.
It should be noted that determinants are associated only with square matrices. If, the matrix is not a square matrix then we cannot find its determinant.
(1) Determinant of 1 x 1 Matrices:
For example we have a 1 x 1 matrix A=
and we want to find value of its determinant. Determinant of 1 x 1 matrix is number itself present in the matrix. Therefore, |A| = 5.
If A =
then |A|=-8
(2) Determinant of 2 x 2 Matrices:
Suppose we have a 2 x 2 matrix of the form
Determinant of such a matrix is equal to (ad-bc).
If,
then |A|=(5 x 10) – (6 x 8)= 50 -48 =2.
Similarly, if
then |A|= (-4 x 11) – (9 x -8) = -44 – (-72) = -44 + 72 = 28
Similarly, we can find determinant of other 2 x 2 matrices.
(3) Determinant of 3 x 3 Matrices:
Determinant of 3 x 3 matrices will be more clear to you when you will learn about co-factors but you can still learn to calculate determinant of 3 x 3 matrices.Suppose we
have a 3 x 3 matrix of the form
Note clearly in the above formula that
|A| =
(determinant of 2 x 2 matrix after eliminating 1st row and 1st column from matrix A) –
(determinant of 2 x 2 matrix after eliminating 1st row and second column from matrix A) +
(determinant of 2 x 2 matrix after eliminating 1st row and third column of matrix A)
We used
,
and
elements (Ist row) to find determinant of matrix A. But, we can use any row or column of matrix A to find value of its determinant. Value of determinant will be same in all the cases. It is better to choose a row or column which has maximum number of zeros. I will explain this in a minute.
Lets suppose we use 3rd row (elements
,
and
) to find value of determinant.
Then,
|A|=
(determinant of 2 x 2 matrix after eliminating 3rd row and first column of matrix A)-
(determinant of 2 x 2 matrix after eliminating 3rd row and second column of matrix A) +
(determinant of 2 x 2 matrix after eliminating 3rd row and 3rd column of matrix A)
=
(
) –
(
) +
(
)
Now, lets take an example of matrix
=5{-40-(-54)} -2 {10-(-36)} + 6(6-(-16) = 5 (-40+54) -2 (10+36) + 6(6+16) = 5(14)-2(46)+6(22) = 70-92+132=110
or
= 4 {-18-(-24)} -6 (-45-6) + 10(-20-2) = 4(-18+24) -6(-51) + 10(-22) = 4(6) + 306 -220 = 110
Note that in both of the cases value of |A| comes out to be same. Similarly, we can find value of determinant using any other row or column.
If, we have
then, it is better to use first column to find determinant because it has maximum number of zeros. The calculation work will be a lot lesser.
|A|=
(determinant of 2 x 2 matrix after eliminating 1st row and first column of matrix A)-
(determinant of 2 x 2 matrix after eliminating 2nd row and 1st column of matrix A) +
(determinant of 2 x 2 matrix after eliminating 3rd row and 1st column of matrix A)
[…] we have any given matrix A and we multiply any row or column of matrix with constant k then determinant of modified matrix becomes k times of determinant of A. We can show this by taking example of 3 x 3 […]