After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.
I hope someone can explain this to me in plain English.
After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.
I hope someone can explain this to me in plain English.
For an $n\times n$ matrix, each of the following is equivalent to the condition of the matrix having determinant $0$:
The columns of the matrix are dependent vectors in $\mathbb R^n$
The rows of the matrix are dependent vectors in $\mathbb R^n$
The matrix is not invertible.
The volume of the parallelepiped determined by the column vectors of the matrix is $0$.
The volume of the parallelepiped determined by the row vectors of the matrix is $0$.
The system of homogenous linear equations represented by the matrix has a non-trivial solution.
The determinant of the linear transformation determined by the matrix is $0$.
The free coefficient in the characteristic polynomial of the matrix is $0$.
Depending on the definition of the determinant you saw, proving each equivalence can be more or less hard.
For me, this is the most intuitive video on the web that explains determinants, and everyone who wants a deep and visual understanding of this topic should watch it:
The determinant by 3Blue1Brown
The whole playlist is available at this link:
Essence of linear algebra by 3Blue1Brown
The crucial part of the series is "Linear transformations and matrices". If you understand that well, everything else will be like a piece of cake. Literally: plain English + visual.
If the determinant of a square matrix $n\times n$ $A$ is zero, then $A$ is not invertible. This is a crucial test that helps determine whether a square matrix is invertible, i.e., if the matrix has an inverse. When it does have an inverse, it allows us to find a unique solution, e.g., to the equation $Ax = b$ given some vector $b$.
When the determinant of a matrix is zero, the system of equations associated with it is linearly dependent; that is, if the determinant of a matrix is zero, at least one row of such a matrix is a scalar multiple of another.
[When the determinant of a matrix is nonzero, the linear system it represents is linearly independent.]
When the determinant of a matrix is zero, its rows are linearly dependent vectors, and its columns are linearly dependent vectors.
The determinant of a matrix is the oriented volume of the image of the unit cube. If it is zero, the unit cube gets mapped inside of a plane and has volume zero.
Let $A$ be a matrix.
Then, $$A^{-1} = \dfrac{1}{detA} adjA$$
Hence, if determinant is zero, Inverse doesn't exist for that matrix.
There is a geometrical interpretation for the determinant. In addition to the interpretation in the another answer, yet another appealing one related to the determinant is its interpretation as the volume of a $N$ dimensional parallelopipped. This is more expressed in 3-Dimensions. If you take $3$ vectors in 3-D, they may or may not form the corners of a parallelopipped, if you take the determinant of a matrix with this 3 vectors as the columns (or rows), if the determinant is zero, it means, they don't form a parallelopipped together, if it is non-zero, it means, they indeed form the 3 edges of a parallelopipped with the volume given by the determinant. The sign of the value of determinant gives a kind of information on the orientation of this body.
When you think a matrix as transformation, the determinant is the area(or volume in higher dimension) made by basis after transformation. This type of thinking will give you visual aid. For example for $n=3$, determinant of a matrix equals zero means, the map by that matrix transforms a plane into a line(the area made by the basis in transformed plane is zero). The idea can be generalized for higher dimension.
Note that, it is not possible to map back a line to a plane. So, inverse of a matrix, which has determinant zero, doesn't exists.
Take a 2 x 2 matrix, call it A, plot that in a coordinate system.
A= [[2,1],[4,2]] . --> Numpy notation of a matrix
Following two vectors are written from A
x=[2,4]
y=[1,2]
If you plot that, you can see that they are in the same span. That means x and y vectors do not form an area. Hence, the det(A) is zero. Det refers to the area formed by the vectors.
Another way of putting it:
If you take $2$ vectors in $2D$ space, you can show that the area of the parallelogram formed is simply the determinant of the matrix formed by those two vectors. This is a general result for $n$-dimensions - the determinant of a matrix is the volume of the $n$-parallelogram formed by the rows of the matrix.
If the determinant is zero, this means the volume is zero. This can only happen when one of the vectors "overlaps" one of the others or more formally, when two of the vectors or linearly dependent.
If a matrix A has an inverse, it means that the matrix multiplication AB = I, the identity matrix. In plain English then, if a matrix is invertible then it may have a solution.
If a matrix's determinant is nonzero, the matrix may have a solution. If the determinant is zero, then the matrix is not invertible and thus does not have a solution because one of the rows can be eliminated by matrix substitution of another row in the matrix.
Common reasons for matrix invertibility are that one or more rows in the matrix is a scalar of the other.
EXAMPLE for a 3x3 matrix: Row 1: x + y + z = P versus Row 2: Qx + Qy + Qz = QP. Row two can be rewritten as Q*(x + y + z = P) or Q*(Row 1)]
OR one of the rows can be created by addition of two or more of the other rows.
EXAMPLE for a 3x3 matrix: Row 1: x + 2y + 3z = P, Row 2: 4x + 5y + 6z = Q, Row 3: 7x + 8y + 9z = P+Q. You see that Row 3 can be duplicated by adding Row 1 and Row 2.
In short, if the determinant of a matrix is zero, the matrix does not have a solution because the matrix cannot be inverted.