Determinate of a matrix

• nlsherrill
In summary, the problem asks to find the determinant of the transpose of a matrix A, where the 2nd and 3rd columns have been swapped. Since each swap of adjacent rows or columns introduces a minus sign in the determinant, the answer is -10.

Homework Statement

From "Introduction to Linear Algebra with applications" by Defranza. Ch.1 section 1.6 prob 30.

Let the matrix

a b c
A = d e f
g h i

Where det(A)= 10

Find

a g d
det b h e
c i f

(sorry I didnt see how to write a matrix in latex, but it should be pretty clear what I mean)

**The matrix looks OK when I am typing it out, but when I submitted the thread it messed with it and it looks really messed up**

Its basically a 3x3 matrix with elements(from a11 to a33) a, b, c, d, e, f, g, h, i

The Attempt at a Solution

So I noticed that the matrix they are asking to find is the transpose of A, plus the 3rd column has been swapped with the 2nd. I haven't read anything about column swapping, so I am not sure if that would alter the the determinate of a matrix like a row swap would. I know det(Atranspose)=det(A), so I figure the answer to this is either 10 or -10(due to the column swap)

Any advice/hints would be appreciated. I also searched in my book and on google and did not find any conclusive results on column swapping as far as elementary row operations go.

What operation on the original matrix corresponds to the column swap on the transpose?

fzero said:
What operation on the original matrix corresponds to the column swap on the transpose?

Ah HA!

so the answer must be -10 correct?

1. What is the determinate of a matrix?

The determinate of a matrix is a numerical value that represents certain properties of the matrix, such as its size, shape, and orientation. It is often denoted as det(A) or |A| and is calculated using a specific formula.

2. How is the determinate of a matrix calculated?

The determinate of a matrix is calculated by finding the sum of the products of the elements in each row or column, depending on the method used. For example, the Laplace expansion method involves finding the determinate of a smaller matrix formed by removing one row and one column, multiplying it by the corresponding element in the original matrix, and repeating this process for each element in the row or column being used.

3. What does the determinate of a matrix tell us?

The determinate of a matrix tells us about certain properties of the matrix, such as whether it is invertible or singular. A non-zero determinate means the matrix is invertible, while a zero determinate means it is singular and has no inverse. It can also tell us about the size and shape of the matrix, as well as its orientation in space.

4. Can the determinate of a matrix be negative?

Yes, the determinate of a matrix can be negative, positive, or zero. This depends on the values of the elements in the matrix and the method used to calculate the determinate. It is important to note that the absolute value of the determinate is what matters, as it represents the magnitude of the matrix's properties.

5. What is the significance of the determinate of a matrix in real-world applications?

The determinate of a matrix is used in various fields, such as engineering, physics, and economics, to solve systems of equations, calculate areas and volumes, and determine the stability of systems. It is also used in computer graphics to transform and manipulate images and in cryptography to encrypt and decrypt messages.