Practice Problems on Properties of Determinants

1. Calculate the determinant of the following matrix and state how row swapping affects the determinant:

[Tex]\begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} [/Tex]

Swap the rows and then calculate the determinant again.

2. Determine the determinant of the matrix below. Then, multiply the first row by 3 and find the new determinant:

[Tex]\begin{vmatrix} 1 & -1 \\ 2 & 3 \end{vmatrix} [/Tex]

Compare the original determinant with the new one to explain the effect of scalar multiplication on the determinant.

3. Calculate the determinant before and after performing a row operation where you add twice the first row to the second row:

[Tex]\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} [/Tex]

4. Calculate the determinant of the following triangular matrix:

[Tex]\begin{vmatrix} 5 & 0 & 0 \\ -1 & 3 & 0 \\ 2 & -2 & 4 \end{vmatrix} [/Tex]

Discuss why the determinant of a triangular matrix is the product of its diagonal elements.

5. Find the determinant of the matrix below, noting what happens when rows are proportional:

[Tex]\begin{vmatrix} 3 & 6 \\ 6 & 12 \end{vmatrix} [/Tex]

Properties of Determinants are the properties that are required to solve various problems in Matrices. There are various properties of the determinant that are based on the elements, rows, and columns of the determinant. These properties help us to easily find the value of the determinant. Suppose we have a matrix M = [a_ij] then the determinant of the matrix is denoted as, |M| or det M.

There are various properties of the determinant of a matrix, and some of the important ones are, Reflection Property, Switching Property, Scalar Multiple Properties, Sum Property, Invariance Property, Factor Property, Triangle Property, Co-Factor Matrix Property, All-Zero Property, and Proportionality or Repetition Property.

In this article, we will learn about all the properties of the determinant with examples and others in detail.