Invertible matrix

In linear algebra, an invertible matrix is a special kind of square matrix that has something called an inverse. This means you can multiply the matrix by its inverse and get back the identity matrix, which acts like the number 1 in matrix multiplication. When a matrix is invertible, it means that the changes it makes can be undone by using its inverse. This idea is very important in many areas of math and science because it helps solve equations and understand how different things are related to each other. Invertible matrices are the same size as their inverses, just like how a key fits perfectly into its lock.

Definition

A square matrix is called invertible if we can find another matrix that, when multiplied together, results in the identity matrix. The identity matrix is a special square matrix where the numbers along the main diagonal are all 1, and all other numbers are 0. When you multiply an invertible matrix by its special matching matrix, called the inverse, you always get the identity matrix back. Finding this inverse matrix is called matrix inversion.

Examples

Here is an example of a special kind of number grid, called a matrix, that can be turned around or "flipped." Imagine a small grid with two rows and two columns:

Matrix A looks like this:
-1 3/2
1 -1

This matrix can be flipped because there is another grid, called Matrix B, that when multiplied together, they make a special grid called the identity matrix, which looks like this:

1 0
0 1

Multiplying Matrix A and Matrix B gives us this special identity grid, which shows that Matrix A can indeed be flipped.

Now, here is another grid that cannot be flipped:

Matrix C looks like this:
2 4
2 4

This grid cannot be flipped because when we try to flip it, we get zero, which means it does not work. This shows us that not all grids can be turned around or flipped.

Methods of matrix inversion

Gaussian elimination is a straightforward way to find the inverse of a matrix. This method involves creating an augmented matrix, where the original matrix is placed next to an identity matrix. By using row operations to change the original matrix into the identity matrix, the identity matrix transforms into the inverse of the original.

Other methods include Newton's method, which uses an iterative approach to approximate the inverse, and the Cayley–Hamilton method, which expresses the inverse using the matrix's determinant and traces. Each method has its uses depending on the size and nature of the matrix involved.

[ A B C D ] − 1 = [ A − 1 + A − 1 B ( M / A ) − 1 C A − 1 − A − 1 B ( M / A ) − 1 − ( M / A ) − 1 C A − 1 ( M / A ) − 1 ] , {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \ (\mathbf {M} /\mathbf {A} )^{-1}\mathbf {CA} ^{-1}&-\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {M} /\mathbf {A} \right)^{-1}\\-\left(\mathbf {M} /\mathbf {A} \right)^{-1}\mathbf {CA} ^{-1}&\left(\mathbf {M} /\mathbf {A} \right)^{-1}\end{bmatrix}},}

[ A B C D ] − 1 = [ ( M / D ) − 1 − ( M / D ) − 1 B D − 1 − D − 1 C ( M / D ) − 1 D − 1 + D − 1 C ( M / D ) − 1 B D − 1 ] . {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {M} /\mathbf {D} \right)^{-1}&-\left(\mathbf {M} /\mathbf {D} \right)^{-1}\mathbf {BD} ^{-1}\\-\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {M} /\mathbf {D} \right)^{-1}&\quad \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {M} /\mathbf {D} \right)^{-1}\mathbf {BD} ^{-1}\end{bmatrix}}.}

( A − B D − 1 C ) − 1 = A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 ( A − B D − 1 C ) − 1 B D − 1 = A − 1 B ( D − C A − 1 B ) − 1 D − 1 C ( A − B D − 1 C ) − 1 = ( D − C A − 1 B ) − 1 C A − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 = ( D − C A − 1 B ) − 1 {\displaystyle {\begin{aligned}\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\\\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\end{aligned}}}

[ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 0 0 ( D − C A − 1 B ) − 1 ] [ I − B D − 1 − C A − 1 I ] . {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {A} -\mathbf {B} \mathbf {D} ^{-1}\mathbf {C} \right)^{-1}&\mathbf {0} \\\mathbf {0} &\left(\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {B} \right)^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {I} &-\mathbf {B} \mathbf {D} ^{-1}\\-\mathbf {C} \mathbf {A} ^{-1}&\mathbf {I} \end{bmatrix}}.}

[ A C T C D ] − 1 = [ A − 1 + A − 1 C T S − 1 C A − 1 − A − 1 C T S − 1 − S − 1 C A − 1 S − 1 ] , {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {C} ^{T}\\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&-\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\\-\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&\mathbf {S} ^{-1}\end{bmatrix}},}

Properties

A square matrix is invertible if it can be multiplied by another matrix to give a special matrix called the identity matrix. This means that invertible matrices have special patterns that allow them to "undo" each other when multiplied together.

When a matrix is invertible, many useful properties follow. For example, its transpose (flipped version) is also invertible. Also, if you know the determinant (a special number calculated from the matrix), an invertible matrix will have a determinant that is not zero. These properties help mathematicians and scientists solve complex problems using matrices.

Derivative of the matrix inverse

Suppose we have a special kind of math square called an invertible matrix, which we’ll call A, and it changes a little bit depending on a number called t. We can find out how the upside-down version of A (called A⁻¹) changes by using a neat math rule.

When A changes a tiny amount, the upside-down version changes in a way we can calculate. This helps us understand how these special squares behave when they shift slightly.

Generalizations

Non-square matrices, which are matrices where the number of rows and columns are different, do not have a regular inverse. However, sometimes they can have a left inverse or a right inverse.

In abstract algebra, these ideas about matrices can be used with different types of numbers and structures, not just the usual real or complex numbers. For some special kinds of structures, the rules for what makes a matrix invertible change. The invertible matrices form a group under matrix multiplication, called the general linear group.

Applications

For most practical uses, you don't need to flip a matrix to solve problems with straight lines. But, for a clear answer, the matrix must be flippable.

Flipping matrices is important in computer graphics, especially for making 3D pictures and games. It helps change how objects look on the screen and simulate real-world actions.

Flipping matrices is also key in MIMO wireless technology. MIMO uses many antennas to send and receive signals at the same time. The matrix must be flippable so the receiver can understand the messages being sent.