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 invertible if we can find another matrix. When we multiply these two matrices together, the result is the identity matrix. The identity matrix is special. It has the number 1 along its main diagonal and 0 everywhere else. When you multiply an invertible matrix by its 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 simple way to find the inverse of a matrix. This method uses an augmented matrix, where the original matrix is placed next to an identity matrix. By changing the original matrix into the identity matrix using row operations, the identity matrix becomes the inverse of the original.

Other methods include Newton's method, which uses steps to estimate the inverse, and the Cayley–Hamilton method, which uses the matrix's determinant and traces to find the inverse. Each method is useful for different sizes and types of matrices.

[ 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 make a special matrix called the identity matrix. This means invertible matrices have special patterns that let them "undo" each other when multiplied.

When a matrix is invertible, it has useful properties. For example, its transpose (flipped version) is also invertible. Also, if you know the determinant (a special number from the matrix), an invertible matrix will have a determinant that is not zero. These properties help mathematicians and scientists solve hard 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

Not all matrices can be inverted. Matrices that are not square—where the number of rows and columns are different—usually do not have a normal inverse. But sometimes, they can have a left inverse or a right inverse.

In a part of math called abstract algebra, these ideas about matrices work with many types of numbers, not just regular real or complex numbers. For some special kinds of number systems, the rules for what makes a matrix invertible are different. The invertible matrices make up a group under matrix multiplication, known as the general linear group.

Applications

You usually don’t need to flip a matrix to solve problems with straight lines. But for a clear answer, the matrix must be able to flip.

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

Flipping matrices is also important in MIMO wireless technology. MIMO uses many antennas to send and receive signals together. The matrix must be able to flip so the receiver can understand the messages.