Linearly Independent Determinant

In mathematics, determinants are linearly independent if none of the determinants can be obtained from the others. When determinants are linearly independent, then each determinant contains new information about the variables. For example

A = [[0],[0],[2]] B = [[2],[-4],[2]] C = [[0],[4],[-4]] D = [[8],[4],[3]] .

Here, the determinants A, B and C are linearly independent but the determinant of D is not since determinant of D is equals to 9A + 4B + 5C. Following example and practice problems will help you to study about linearly independent determinant.
Example Problems of Linearly Independent Determinant:

Example problem 1:

Show that the determinant [[2,3],[4,5]] is linearly dependent or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

[[2,3],[4,5]] .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

[[2,3],[4,5]] = (2 * 5) - (3 * 4)

= -2

Since the value of the determinant is not equal to 0, the given determinant is linearly independent.

Step 4: Solution

Hence, the given determinant [[2,3],[4,5]] is linearly independent.

Example problem 2:

Show that the determinant [[1,4],[5,20]] is linearly dependent or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

[[1,4],[5,20]] .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

[[1,4],[5,20]] = (1 * 20) - (4 * 5)

= 0

Since the value of the determinant is equal to 0, the given determinant is linearly dependent.

Step 4: Solution

Hence, the given determinant [[1,4],[5,20]] is linearly dependent.
Practice Problems of Linearly Independent Determinant:

1) Show that the determinant [[4,3],[2,8]] is linearly dependent or not using Wronskian determinant method.

2) Show that the determinant [[2,5],[4,10]] is linearly dependent or not using Wronskian determinant method.

Solutions:

1) The given determinant [[4,3],[2,8]] is linearly independent.

2) The given determinant [[2,5],[4,10]] is linearly dependent.