# How can a determinant of a matrix become an area?

• RyozKidz
In summary, the conversation discusses the concept of determinants of matrices and how they relate to areas and volumes. It is explained that a matrix can map squares to parallelograms or cubes to parallelipipeds, and the determinant is the ratio of the new shape's area or volume to the original shape's. The conversation also mentions that determinants can be defined by considering how row operations affect volumes, but it is more common for courses to start with the determinant formula and then show its relationship to volume factors.
RyozKidz
can anyone explain or prove this??

Ax={(xTAT)T}

how can a determinant of a matrix become an area??

example: 2 X 2 matrix

the determinant of this matrix is ad-bc !
but i search on wikipedia it wrote like this :The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors [0,0] [1,0] [1,1] [0,1]in turn. These row vectors define the vertices of the unit square.

The action of a matrix on vectors in the plane is to map squares to parallelograms. The parallelograms may be skewed, rotated, scaled, flipped, etc by the matrix. The determinant is the ratio of the new parallelogram area to the original square area. If the square gets flipped or turned "inside out", then it is as if the parallelogram has "negative" area, so the determinant is given a minus sign.

In 3 dimension, a matrix maps cubes to parallelipipeds, and the determinant is the ratio of the volumes.

You can actually define determinants in this way, think about how swapping and scaling rows affects volumes, and then derrive the determinant formula via row operations. However most courses go the other way around and start off with the formula, then show it is the volume factor.

## 1. How is a determinant of a matrix related to area?

The determinant of a 2x2 matrix represents the area of the parallelogram formed by the two column vectors of the matrix. In a 3x3 matrix, the determinant represents the volume of the parallelepiped formed by the three column vectors.

## 2. What is the formula for calculating the determinant of a matrix?

The formula for calculating the determinant of a 2x2 matrix is ad-bc, where a, b, c, and d are the elements of the matrix. For a 3x3 matrix, the formula is a(ei-fh) - b(di-fg) + c(dh-eg), where a, b, c, d, e, f, g, and h are the elements of the matrix.

## 3. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. In a 2x2 matrix, a negative determinant indicates that the two column vectors are oriented in opposite directions, resulting in a negative area. In a 3x3 matrix, a negative determinant indicates that the three column vectors are oriented in a way that results in a negative volume.

## 4. What does a determinant of 0 mean?

A determinant of 0 means that the two or three column vectors of the matrix are linearly dependent, resulting in an area or volume of 0. This can also mean that the matrix is not invertible.

## 5. How is the determinant of a matrix used in real-world applications?

The determinant of a matrix is used in various fields such as physics, engineering, and economics to solve systems of equations and determine the properties of a system. It is also used in computer graphics to calculate transformations and rotations of objects.

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