Invertible Matrix - Theorems, Properties, Definition, Examples

Invertible Matrix

In linear algebra, an n-by-n square matrix is called invertible (also non-singular or non-degenerate), if the product of the matrix and its inverse is the identity matrix. In other words, an invertible matrix is a matrix for which the inverse can be calculated. Let us learn the conditions for a given matrix to be invertible and theorems associated with the invertible matrix and their proofs.

1.	What is Invertible Matrix?
2.	Invertible Matrix Theorem
3.	Invertible Matrix Properties
4.	Invertible Matrix Determinant
5.	Applications of Invertible Matrix
6.	FAQs on Invertible Matrix

What is Invertible Matrix?

An invertible matrix is a matrix for which matrix inversion operation exists, given that it satisfies the requisite conditions. Any given square matrix A of order n × n is called invertible if there exists another n × n square matrix B such that, AB = BA = I\(_n\), where I\(_n\) is an identity matrix of order n × n.

Invertible Matrix Example

The examples of an invertible matrix are given below. It can be observed that the determinant of these matrices is non-zero.

• Matix A = \(\left[\begin{array}{ccc} 2 & 7 \\ \\ 2 & 8 \end{array}\right]\) is a 2 × 2 invertible matrix as det A = 2(8) - 2(7) = 16 - 14 = 2 ≠ 0.
• Matrix B = \(\left[\begin{array}{ccc} 1 & -4 & 2 \\ -2 & 1 & 3 \\ 2 & 6 & 8 \end{array}\right]\) is a 3 × 3 invertible matrix as det A = 1 (8 - 18) + 4 (-16 - 6) + 2(-12 - 2) = -126 ≠ 0.

Invertible Matrix Theorem

The invertible matrix theorem is a theorem in linear algebra which offers a list of equivalent conditions for an n×n square matrix A to have an inverse. Any square matrix A over a field R is invertible if and only if any of the following equivalent conditions(and hence, all) hold true.

• A is row-equivalent to the n × n identity matrix I\(_n\).
• A is column-equivalent to the n-by-n identity matrix I\(_n\).
• A is invertible, that is, A has an inverse and A is non-singular or non-degenerate.
• The determinant of A is not zero.
• There is an n-by-n square matrix B such that AB = I\(_n\) = BA.
• Matrix A has 'n' pivot positions.
• The equation Ax = 0 has only trivial solution given as, x = 0.
• The columns of matrix A form a linearly independent set.
• The columns of A span Rn.
• For each column vector b in Rn, the equation Ax = b has a unique solution.
• There is an n×n matrix M such that MA = I\(_n\).
• There is an n×n matrix N such that AN = I\(_n\).
• The transpose matrix AT is also invertible.
• The columns of A form a basis for Rn.
• The rank of A is n.
• The null space of A is {0}.
• 0 is not an eigenvalue of A.

Invertible Matrix Properties

There are different properties associated with an invertible matrix. Some of these are listed below:

• If A is non-singular, then so is A-1 and (A -1)-1 = A.
• If A and B are non-singular matrices, then AB is non-singular and (AB)-1 = B-1 A-1.
• If A is non-singular then (AT)-1 = (A-1)T.
• If A and B are matrices with AB = I\(_n\) then A and B are inverses of each other. ⇒ AB = I then BA = I. (Let A, A\(_1\), and A\(_2\) be n × n matrices, the following statements are true.)
• If A has an inverse matrix, then there is only one inverse matrix.
• If A\(_1\) and A\(_2\) have inverses, then A\(_1\) A\(_2\) has an inverse and (A\(_1\) A\(_2\))-1 = A\(_2\)-1 A\(_1\)-1
• If A has an inverse, then x = A-1d is the solution of Ax = d and this is the only solution.
• The following are equivalent: (i) A has an inverse. (ii) det (A) is not zero. (iii) Ax = 0 implies x = 0.
• If c is any non-zero scalar then cA is invertible and (cA)-1 = A-1/c.
• det A-1 = (det A)-1

Invertible Matrix Determinant

The invertible matrix determinant is the inverse of the determinant: det(A-1) = 1 / det(A). Let us check the proof of the above statement.

Invertible Matrix Determinant Proof:

We know that, det(A • B) = det (A) × det(B)

Also, A × A-1 = I

⇒ det(A •A-1) = det(I)

or, det(A) × det(A-1) = det(I)

Since, det(I) = 1

⇒det(A) × det(A-1) = 1

or, det(A-1) = 1 / det(A)

Hence, proved.

Applications of Invertible Matrix

Invertible matrices find application in different fields in our day-to-day lives. They are really useful for a variety of things, but they really come into their own for 3D transformations. Here are few applications of invertible matrices,

• Invertible matrices can be used to encrypt a message. There are many ways to encrypt a message and the use of coding has become particularly significant in recent years.
• Invertible matrices are employed by cryptographers to decode a message as well, especially those programming the specific encryption algorithm.
• Computer graphics in the 3D space use invertible matrices to render what you see on the screen.

Invertible Matrix Important Notes:

• The inverse of an invertible matrix is unique.
• If A and B are two invertible matrices of the same order then (AB)-1 = B-1A-1.
• A square matrix A is invertible, only if its determinant is a non-zero value, |A| ≠ 0.

☛Related Topics:

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• Determinant Formula
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• Square Matrix
• Matrix Formula