We would like to find the determinant of

(1)

We write

(2)

We would like to find the determinant of

(3)

We start by using the first row and definition of determinant

(4)

Suppose we are given

(5)

We want to find the value(s) of such that . We know determinant of is computed by

(6)

For this to be equal to zero, we must have

(7)

Which means

(8)

Suppose we are given

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We would like to use elementary row operations to find the determinant of .

1. Interchange the first and second row. We know this will result in

(10)

2. Multiply the first row by and add the first row to the second. This will result in no change in determinant

(11)

3. Next multiply the second row by

(12)

4. Next, multiply the first row by and add to the third row. No change in determinant

(13)

5. Now multiply the second row by and add it to the third row (again no change in determinant)

(14)

We can see now that the determinant of the last obtained matrix is and so

(15)

Suppose we are given the matrix

(16)

We would like to use elementary row operations to find determinant of We start by noticing the following steps:

1. If we multiply the first row of by and obtain

(17)

2. Now multiply the first row by and add the first row to the second row. In this case no change happens in determinant

(18)

3. Next multiply the first row by and add the first row to the last row. Here again no change in determinant

(19)

As you see the matrix obtained which is row equivalent to is an upper triangular matrix and determinant of an upper or lower triangular matrix is the multiplication of the diagonal elements

(20)