Inverse of Matrix

Inverse of Matrix for a matrix A is denoted by A^-1. The inverse of a 2 × 2 matrix can be calculated using a simple formula. Further, to find the inverse of a 3 × 3 matrix, we need to know about the determinant and adjoint of the matrix. The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity.

The inverse of matrix is used of find the solution of linear equations through the matrix inversion method. Here, let us learn about the formula, methods, and terms related to the inverse of matrix.

What is Inverse of Matrix?

The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^-1, and A.A^-1 = A^-1·A = I, where I is the identity matrix. The matrix whose determinant is non-zero and for which the inverse matrix can be calculated is called an invertible matrix. For example, the inverse of A =

But how to find the inverse of a matrix? Let us see in the upcoming sections.

Inverse Matrix Formula

In the case of real numbers, the inverse of any real number a was the number a^-1, such that a times a^-1 equals 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn’t zero. The inverse of a square matrix A, denoted by A^-1, is the matrix so that the product of A and A^-1 is the identity matrix. The identity matrix that results will be the same size as matrix A.

Since |A| is in the denominator of the formula, the inverse of matrix exists only if the determinant of the matrix is a non-zero value. i.e., |A| ≠ 0.

Inverse Matrix Formula in Math

The inverse matrix formula for a matrix A is given as,

• A^-1 = adj(A)/|A|; |A| ≠ 0

where A is a square matrix.

Note: For inverse of a matrix to exist:

• The given matrix should be a square matrix.
• The determinant of the matrix should not be equal to zero.

Terms Related to Inverse of Matrix

The following terms below are helpful for more clear understanding and easy calculation of the inverse of matrix.

Minor: The minor is defined for every element of a matrix. The minor of a particular element is the determinant obtained after eliminating the row

Cofactor: The cofactor of an element is calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements in order representation of that element.

Cofactor of aij = (-1)^{i + j}× minor of aij.

Determinant: The determinant of a matrix is the single unique value representation of a matrix. The determinant of the matrix can be calculated with reference to any row or column of the given matrix. The determinant of the matrix is equal to the summation of the product of the elements and its cofactors, of a particular row or column of the matrix.

Singular Matrix: A matrix having a determinant value of zero is referred to as a singular matrix. For a singular matrix A, |A| = 0. The inverse of a singular matrix does not exist.

Non-Singular Matrix: A matrix whose determinant value is not equal to zero is referred to as a non-singular matrix. For a non-singular matrix |A| ≠ 0. A non-singular matrix is called an invertible matrix since its inverse can be calculated.

Adjoint of Matrix: The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.

Rules For Row and Column Operations of a Determinant: The following rules are helpful to perform the row and column operations on determinants.

• The value of the determinant remains unchanged if the rows and columns are interchanged.
• The sign of the determinant changes, if any two rows or (two columns) are interchanged.
• If any two rows or columns of a matrix are equal, then the value of the determinant is zero.
• If every element of a particular row or column is multiplied by a constant, then the value of the determinant also gets multiplied by the constant.
• If the elements of a row or a column are expressed as a sum of elements, then the determinant can be expressed as a sum of determinants.
• If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.

Methods to Find Inverse of Matrix

The inverse of matrix can be found using two methods. The inverse of a matrix can be calculated through elementary operations and through the use of an adjoint of a matrix. The elementary operations on a matrix can be performed through row or column transformations. Also, the inverse of a matrix can be calculated by applying the inverse of matrix formula through the use of the determinant and the adjoint of the matrix. For performing the inverse of the matrix through elementary column operations we use the matrix X and the second matrix B on the right-hand side of the equation.

• Elementary row or column operations
• Inverse of matrix formula(using the adjoint and determinant of matrix)

Let us check each of the methods described below.

Elementary Row Operations

To calculate the inverse of matrix A using elementary row transformations, we first take the augmented matrix [A | I], where I is the identity matrix whose order is the same as A. Then we apply the ropw operations to convert the left side A into I. Then the matrix gets converted into [I | A^-1]. For a more detailed process

Elementary Column Operations

We can apply the column operations as well just like how the process was explained for row operations to find the inverse of matrix.

Inverse of Matrix Formula

The inverse of matrix A can be computed using the inverse of matrix formula, by dividing the adjoint of a matrix by the determinant of the matrix. The inverse of a matrix can be calculated by following the given steps:

• Step 1: Calculate the minors of all elements of A.
• Step 2: Then compute cofactors of all elements and write the cofactor matrix by replacing the elements of A by their corresponding cofactors.
• Step 3: Find the adjoint of A (written as adj A) by taking the tranpose of cofactor matrix of A.
• Step 4: Multiply adj A by reciprocal of determinant.

In this section, we have learned the different methods to calculate the inverse of a matrix. Let us understand it better using a few examples for the different orders of matrices in the “examples” section below.

Inverse of 2 × 2 Matrix

The inverse of 2 × 2 matrix is easier to calculate in comparison to matrices of higher order. We can calculate the inverse of 2 × 2 matrix using the general steps to calculate the inverse of a matrix. Let us find the inverse of the 2 × 2 matrix given below:

Therefore, in order to calculate the inverse of 2 × 2 matrix, we need to first swap the positions of terms a and d and put negative signs for terms b and c, and finally divide it by the determinant of the matrix.

Inverse of 3 × 3 Matrix

We know that for every non-singular square matrix A, there exists an inverse matrix A^-1, such that A × A^-1 = I. Let us take any 3 × 3 square matrix given as,

The inverse of 3×3 matrix can be calculated using the inverse matrix formula, A^-1 = (1/|A|) × Adj A

We will first check if the given matrix is invertible, i.e., |A| ≠ 0. If the inverse of matrix exists, we can find the adjoint of the given matrix and divide it by the determinant of the matrix.

The similar method can be followed to find the inverse of any n × n matrix. Let us see if similar steps can be used to calculate the inverse of m × n matrix.

Inverse of 2 × 3 Matrix

We know that the first condition for the inverse of a matrix to exist is that the given matrix should be a square matrix. Also, the determinant of this square matrix should be non-zero. This means that the inverse of matrices of the order m × n will not exist where m ≠ n. Therefore, we cannot calculate the inverse of 2 × 3 matrix.

Inverse of 2 × 1 Matrix

Similar to the inverse of 2 × 3 matrix, the inverse of 2 × 1 matrix will also not exist because the given matrix is not a square matrix.

Determinant of Inverse Matrix

The determinant of the inverse of an invertible matrix is the inverse of the determinant of the original matrix. i.e., det(A^-1) = 1 / det(A). Let us check the proof of the above statement.

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

Important Points on Inverse of a Matrix:

The following points are helpful to understand more clearly the idea of the inverse of matrix.

• The inverse of a square matrix if exists, is unique.
• If A and B are two invertible matrices of the same order then (AB)^-1 = B^-1A^-1.
• The inverse of a square matrix A exists, only if its determinant is a non-zero value, |A| ≠ 0.
• The elements of a row or column, if multiplied with the cofactor elements of any other row or column, then their sum is zero.
• The determinant of the product of two matrices is equal to the product of the determinants of the two individual matrices. |AB| = |A|.|B|

Let us see how to use the inverse matrix formula in the following solved examples section.

Solved Examples on Inverse of Matrix

Now, we will determine the adjoint of the matrix A by calculating the cofactors of each element and then taking the transpose of the cofactor matrix.

Identity Matrix

We just mentioned the “Identity Matrix”. It is the matrix equivalent of the number “1”:

• It is “square” (has same number of rows as columns),
• It has 1s on the diagonal and 0s everywhere else.
• Its symbol is the capital letter I.

The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc …

Definition

Here is the definition:

The inverse of A is A^-1 only when:

AA^-1 = A^-1A = I

Sometimes there is no inverse at all.

(Note: writing AA^-1 means A times A^-1)

2×2 Matrix

OK, how do we calculate the inverse?

Well, for a 2×2 matrix the inverse is:

How do we know this is the right answer?

Remember it must be true that: AA^-1 = I

So, let us check to see what happens when we multiply the matrix by its inverse:

It should also be true that: A^-1A = I

Why don’t you have a go at multiplying these? See if you also get the Identity Matrix:

Why Do We Need an Inverse?

Because with matrices we don’t divide! Seriously, there is no concept of dividing by a matrix.

But we can multiply by an inverse, which achieves the same thing.

Imagine we can’t divide by numbers …

… and someone asks “How do I share 10 apples with 2 people?”

But we can take the reciprocal of 2 (which is 0.5), so we answer:

10 × 0.5 = 5

They get 5 apples each.

The same thing can be done with matrices:

Say we want to find matrix X, and we know matrix A and B:

XA = B

It would be nice to divide both sides by A (to get X=B/A), but remember we can’t divide.

But what if we multiply both sides by A^-1 ?

XAA^-1 = BA^-1

And we know that AA^-1 = I, so:

XI = BA^-1

We can remove I (for the same reason we can remove “1” from 1x = ab for numbers):

X = BA^-1

And we have our answer (assuming we can calculate A^-1)

In that example we were very careful to get the multiplications correct, because with matrices the order of multiplication matters. AB is almost never equal to BA.

A Real Life Example: Bus and Train

The answer almost appears like magic. But it is based on good mathematics.

Calculations like that (but using much larger matrices) help Engineers design buildings, are used in video games and computer animations to make things look 3-dimensional, and many other places.

It is also a way to solve Systems of Linear Equations.

The calculations are done by computer, but the people must understand the formulas.

Order is Important

Say that we are trying to find “X” in this case:

AX = B

This is different to the example above! X is now after A.

With matrices the order of multiplication usually changes the answer. Do not assume that AB = BA, it is almost never true.

So how do we solve this one? Using the same method, but put A^-1 in front:

A^-1AX = A^-1B

And we know that A^-1A= I, so:

IX = A^-1B

We can remove I:

X = A^-1B

And we have our answer (assuming we can calculate A^-1)

Why don’t we try our bus and train example, but with the data set up that way around.

It can be done that way, but we must be careful how we set it up.

This is what it looks like as AX = B:

Same answer: 16 children and 22 adults.

So matrices are powerful things, but they do need to be set up correctly!

The Inverse May Not Exist

First of all, to have an inverse the matrix must be “square” (same number of rows and columns).

But also the determinant cannot be zero (or we end up dividing by zero). How about this:

24−24? That equals 0, and 1/0 is undefined.
We cannot go any further! This matrix has no Inverse.

Such a matrix is called “Singular”,
which only happens when the determinant is zero.

And it makes sense … look at the numbers: the second row is just double the first row, and does not add any new information.

And the determinant 24−24 lets us know this fact.

(Imagine in our bus and train example that the prices on the train were all exactly 50% higher than the bus: so now we can’t figure out any differences between adults and children. There needs to be something to set them apart.)

Conclusion

• The inverse of A is A^-1 only when AA^-1 = A^-1A = I
• To find the inverse of a 2×2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).
• Sometimes there is no inverse at all

Matrix Inverse

Inverse Matrix

Before calculating the inverse of a matrix let us understand what a matrix is? A matrix is a definite collection of objects arranged in rows and columns These objects are called elements of the matrix. The order of a matrix is written as number rows by number of columns. For example, 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4 and so on. We can find the matrix inverse only for square matrices, whose number of rows and columns are equal such as 2 × 2, 3 × 3, etc. In simple words, inverse matrix is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix. In this article, you will learn what a matrix inverse is, how to find the inverse of a matrix using different methods, properties of inverse matrix and examples in detail.

Matrix Inverse

If A is a non-singular square matrix, there is an existence of n x n matrix A^-1, which is called the inverse matrix of A such that it satisfies the property:

AA^-1 = A^-1A = I, where I is  the Identity matrix

The identity matrix for the 2 x 2 matrix is given by

It is noted that in order to find the inverse matrix, the square matrix should be non-singular whose determinant value does not equals to zero.

Let us take the square matrix A

Where a, b, c, and d represents the number.

The determinant of the matrix A is written as ad-bc, where the value of determinant should not equal to zero for the existence of inverse. The inverse matrix can be found for 2× 2, 3× 3, …n × n matrices. Finding the inverse of a 3×3 matrix is a bit more difficult than finding the inverses of a 2 ×2 matrix.

Inverse Matrix Method

The inverse of a matrix  can be found using the three different methods. However, any of these three methods will produce the same result.
Method 1:

Similarly, we can find the inverse of a 3×3 matrix by finding the determinant value of the given matrix.

Method 2:

One of the most important methods of finding the matrix inverse involves finding the minors and cofactors of elements of the given matrix. Observe the below steps to understand this method clearly.

• The inverse matrix is also found using the following equation:

A^-1= adj(A)/det(A),

          where adj(A) refers to the adjoint of a matrix A, det(A) refers to the determinant of a matrix A.

• The adjoint of a matrix A or adj(A) can be found using the following method.

          In order to find the adjoint of a matrix A first, find the cofactor matrix of a given matrix and then   

          take the transpose of a cofactor matrix.

• The cofactor of a matrix can be obtained as

C_ij = (-1)^ij det (Mij)

Here, Mij refers to the (i,j)^th minor matrix after removing the ith row and the jth column. You can also say that the transpose of a cofactor matrix is also called the adjoint of a matrix A.

Similarly, we can also find the inverse of a 3 x 3 matrix. Here also the first step would be to find the determinant, followed by the next step – Transpose.
Method 3:

Finding an Inverse Matrix by Elementary Transformation

Let us consider three matrices X, A and B such that X = AB. To determine the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix. Learn more about  how to do elementary transformations of matrices here.

If the inverse of matrix A, A^-1 exists then to determine A^-1 using elementary row operations

1. Write A = IA, where I is the identity matrix of the same order as A.
2. Apply a sequence of row operations till we get an identity matrix on the LHS and use the same elementary operations on the RHS to get I = BA. The matrix B on the RHS is the inverse of matrix A.
3. To find the inverse of A using column operations, write A = IA and apply column operations sequentially till I = AB is obtained, where B is the inverse matrix of A.

Inverse of a Matrix Formula

Inverse Matrix 2 x 2 Example

To understand this concept better let us take a look at the following example.

Example: Find the inverse of matrix A given below:

Inverse Matrix 3 x 3 Example

Problem:

Solution:

Determinant of the given matrix is

Let us find the minors of the given matrix as given below:

Now, find the adjoint of a matrix by taking the transpose of cofactors of the given matrix.

Frequently Asked Questions – FAQs

What is concept inverse of a matrix?

Like numbers, matrices do have reciprocals. In case of matrices, this reciprocal is called inverse matrix. If A is a square matrix and B is its inverse, then the product of two matrices is equal to the unit matrix.

How do you find the inverse of a 3×3 matrix?

The steps required to find the inverse of a 3×3 matrix are:
Compute the determinant of the given matrix and check whether the matrix invertible
Calculate the determinant of 2×2 minor matrices
Formulate the matrix of cofactors
Take the transpose of the cofactor matrix to get the adjugate matrix
Finally, divide each term of the adjugate matrix by the determinant

Is adjoint and inverse the same?

No, adjoint matrix and inverse matrix are not the same. However, by dividing the each term of the adjugate matrix by the determinant of the original matrix, we get inverse matrix.

How to do you know whether the given matrix has inverse?

If the determinant of a given matrix is not equal to 0, then the matrix is invertible and we can find the inverse of such matrix. That means, the given matrix must be non-singular.

What are the properties of inverse matrix?

Some of the important properties of inverse matrices are:
The inverse of inverse matrix is equal to the original matrix.
If A and B are invertible matrices, then AB is also invertible. Thus, (AB)^-1 = B^-1A^-1
If A is nonsingular then (A^T)^-1 = (A^-1)^T
The product of a matrix and its inverse and vice versa is always equal to the identity matrix.

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