# Inverse Matrix Formula

## Inverse Matrix Formula

A-1 is an abbreviation for Inverse Matrix for matrix A. There is a straightforward Inverse Matrix Formula for calculating the inverse of a 2×2 matrix. In order to find the inverse of a matrix of order 3 or higher, students also need to be aware of the matrix’s determinant and adjoint. When multiplied by the provided matrix, the other matrix that serves as the inverse of that matrix yields the multiplicative identity.

The matrix inversion method is used to obtain the answer to linear equations by using the Inverse Matrix Formula of the matrix. Even a matrix has an inverse, just like every number has its inverse. If matrix A is taken into account, its inverse is designated as A-1. An additional matrix called the inverse of a matrix is one that, when multiplied by the original matrix, produces the multiplicative identity.

The Inverse Matrix Formula of a matrix A is A-1. And I, the identity matrix, is equal to A. A-1. The square matrix needs to be non-singular and have a non-zero determinant value in order to get the inverse matrix. A matrix is a set of things that are clearly ordered in rows and columns. These things are referred to as “matrix elements.” A matrix’s order is expressed as the product of the number of rows and the number of columns. Using the examples of 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4, and so on. Only square matrices with an equal number of rows and columns, such as 2 × 2, 3 ×  3, etc., can have an Inverse Matrix Formula found. In other words, the adjugate of the supplied matrix is divided by the determinant of the given matrix to produce the Inverse Matrix Formula. A matrix is invertible, and students can get its inverse if its determinant is not equal to 0. If it is not, then the matrix is given. The resulting requirement is that the given matrix not be singular.

If A is a non-singular square matrix, then A-1, often known as the Inverse Matrix Formula of A, must exist if it is to satisfy the following property:

Where I is the Identity matrix, AA-1 = A-1A = I

I=1/0, 0/1 is the identity matrix for the 2 x 2 matrix. It should be emphasised that the square matrix must not be singular and that the determinant value must not equal zero in order to find the Inverse Matrix Formula.

A= a/c, b/d

where the numbers a, b, c, and d are represented.

The determinant of matrix A is denoted by the notation ad-bc, where the presence of an inverse requires that the determinant’s value be greater than zero. For matrices of the form2× 2, 3× 3, …n × n ,one can find the Inverse Matrix Formula. It is slightly more challenging to find the Inverse Matrix Formula of a 3×3  matrix than it is to find the Inverse Matrix Formula of a 2 ×2 matrix.

To comprehend the concept of the Inverse Matrix Formula more clearly, students must consider the following points.

1. If it exists, a square matrix’s inverse is the only one.
2. If A and B are two identically ordered invertible matrices, then (AB)-1 = B-1A-1.
3. If |A| 0, the determinant of a square matrix A must not be zero in order for the inverse to exist.
4. The reciprocal of the determinant of the original matrix is the same as the determinant of the inverse matrix.
5. The product of the determinants of the two individual matrices is equal to the determinant of the product of two matrices. |AB| = |A|.|B|

## What is Inverse of Matrix?

The multiplicative identity is obtained by multiplying the provided matrix by the other matrix that serves as the inverse of the matrix. A matrix’s inverse is A-1, and since I is the identity matrix, A A-1 = A-1 A = I. An invertible matrix is one for which the Inverse Matrix Formula can be determined and whose determinant is non-zero. By dividing a matrix by its Inverse Matrix Formula, the Identity Matrix (I) is produced. One is the value of the matrix called the Identity Matrix. A diagonal matrix known as an identity matrix has all of its diagonal components equal to 1 and all of its other components equal to 0. It is also known as an Elementary Matrix or a Unit Matrix.

The following are some crucial characteristics of inverse matrices:

1. The original matrix is represented by the inverse of the inverse matrix.
2. A and B must also be invertible matrices for AB to be invertible. Consequently, (AB)^-1 = B^-1A^-1
3. If A is not a singular number, then  (A^T)^-1 = (A^-1)^T
4. The identity matrix is always equal to the product of a matrix and its inverse, and vice versa.

Finding the minors and cofactors of the given matrix’s components is one of the most crucial steps in determining its Inverse Matrix Formula. To comprehend this approach clearly, follow the steps listed below.

The following equation can also be used to find the Inverse Matrix Formula:

Adjoint of a matrix A is referred to as adj(A), while the determinant of a matrix A is referred to as det(A).

The method below can be used to determine the adjoint of a matrix A or adj(A).

Find the cofactor matrix of the given matrix to determine the adjoint of matrix A, and then

Consider a cofactor matrix transposed.

The formula Cij = (-1)i+j det (Mij) can be used to determine a matrix’s cofactor (Mij)

The (i,j)th minor matrix is referred to as Mij in this context after the removal of the ith row and jth column. The adjoint of matrix A is another name for the transposition of a cofactor matrix.

Find out how to find a matrix’s adjoint here.

A 3 x 3 Inverse Matrix Formula can also be found in a similar manner. Finding the determinant would be the first step in this case as well, followed by transposition.

## Inverse Matrix Formula

The symbol for matrix A’s inverse is A-1. A matrix A’s inverse is A-1. The identity matrix’s I is also equal to A. A-1. To obtain the inverse matrix, the square matrix must be non-singular and have a non-zero determinant value. The inverse of a 2*2 matrix can be calculated using a simple formula. Students also need to be familiar with the matrix’s determinant and adjoint in order to find the inverse of a matrix of order 3 or higher. The other matrix, acting as its Inverse Matrix Formula produces the multiplicative identity when multiplied by the given matrix. A group of items with a distinct row and column hierarchy is referred to as a matrix.

A group of items with a distinct row and column hierarchy is referred to as a “matrix.” Matrix elements are what these items are called. The ratio of rows to columns is used to indicate the order of a matrix. By employing the Inverse Matrix Formula, the matrix inversion method can be utilised to find the solution to linear equations. Every number has its Inverse Matrix Formula, and even matrices have them. Matrix A’s inverse is known as A-1 when it is taken into consideration. The multiplicative identity is produced when the original matrix is multiplied by an additional matrix, known as the Inverse Matrix Formula.

In the case of real numbers, the number a-1 served as the inverse of any real number a, such that a multiplied by a-1 = 1. As long as the number was not zero, the inverse of a real number was equal to the reciprocal of the number. The identity matrix is the product of a square matrix. A and its inverse, indicated by the symbol A-1. The resulting identity matrix will have the same dimensions as matrix A.

A square matrix’s Inverse Matrix Formula, A-1, exists only when A*A-1= A-1*A=I

The inverse of a matrix can only exist if the matrix’s determinant has a non-zero value because |A| is in the denominator of the expression above. i.e., |A| ≠ 0.

### How to Find Matrix Inverse?

The following formula to determine the inverse of a square matrix A: A-1 = adj(A) / |A|; |A| ≠ 0

where

A is a square matrix.

A’s determinant is |A|.

Observation: For a matrix to have its inverse:

The provided matrix must be square.

It is not acceptable for the matrix’s determinant to equal zero

### Terms Related to Matrix Inverse

The definitions of the terms below will make it easier to understand and calculate the Inverse Matrix Formula of a matrix. Minor: Each component of a matrix has a determined minor. The determinant left over after removing the row and column containing a certain element is its minor. For a matrix A = a11, a12, a13.

Cofactor: To find the cofactor of an element, multiply the minor by -1 and add the exponent of the sum of the row and column elements that make up that element’s representation in sequence.

Determinant: A matrix’s determinant is its unique, distinct value representation. Any row or column of the provided matrix can be used to calculate the matrix’s determinant. A certain row or column of the matrix’s determinant is equal to the sum of the products of the elements and its cofactors.

Singular Matrix: A singular matrix is a matrix with a determinant value of zero. |A| = 0 for a singular matrix A. A singular matrix does not have an Inverse Matrix Formula.

Non-Singular Matrix: A non-singular matrix is one whose determinant value does not equal zero. Since |A| 0 for a non-singular matrix exists, so does its Inverse Matrix Formula.

Matrix Adjoint: The adjoint of a matrix is the cofactor element matrix of the specified matrix transposed.

Rules for a Determinant’s Row and Column Operations:

The row and column operations on determinants can be accomplished with the aid of the following rules.

If the rows and columns are switched, the determinant’s value is unaffected.

If any two rows or (two columns) are switched, the determinant’s sign changes.

Any two equal rows or columns in a matrix result in the determinant’s value being zero.

The value of the determinant is multiplied by the constant if each element in a given row or column is multiplied by the constant.

The determinant can be expressed as a sum of determinants if the elements of a row or column are expressed as a sum of elements.

The value of the determinant is unaffected if the elements of one row or column are added to or

subtracted from the corresponding multiples of elements in another row or column.

### Methods to Find Inverse of Matrix

There are two ways to find a Inverse Matrix Formula. Through the use of an adjoint of a matrix and simple processes, it is possible to derive the inverse of a matrix. Row or column transformations can be used to carry out the basic operations on a matrix. Additionally, the adjoint and determinant of the matrix can be used to apply the inverse of the matrix formula in order to get the Inverse Matrix Formula. Students must use the first matrix X and the second matrix B on the right-hand side of the equation to perform the inverse of the matrix using simple column operations.

simple row- or column-based operations

Inverse Matrix Formula (using the adjoint and determinant of matrix)

### Inverse of 2 x 2 Matrix

The inverse of a 2×2 matrix can be simply calculated using the formula below.

The determinant of a 2×2 matrix A = ⎡⎢⎣abcd⎤⎥⎦ [ a b c d ] is |A| = ad – bc.

The locations of a and d are switched, b and c are given a negative sign, and the matrix is divided by its determinant, ad-bc. Multiply the Inverse Matrix Formula by the original matrix to see if the Inverse Matrix Formula we found is the correct solution. The solution is accurate if the multiplication results in an identity matrix.

It can be easily created by simply cross multiplying the constituent parts, starting from the top left, and then removing the results.

simple row- or column-based operations

Inverse Matrix Formula (using the adjoint and determinant of matrix)

Technique 2: Simple Row/Column Operations

The Gauss-Jordan method is another name for this approach. Using the basic transformation technique, determine the Inverse Matrix Formula by converting it into an identity matrix. The augmented matrix [A | I] or A = I A, in which I is the identity matrix with the same order as A, serves as our starting point. The left side is then transformed from A to I using row/column operations. After that, the matrix is changed into [I | A-1] or I = A-1 A.

The following are some simple row operations we can perform:

change rows.

Each element in a row should be multiplied by or divided by a fixed amount.

A multiple of another row can be added to or subtracted from a row to replace it.

Method 3- Using the adjoint matrix and the determinant

Inverse A-1 = (1/|A|) x Adj A for matrix A.

Note: To determine whether the matrix is non-singular and invertible, first determine whether |A| 0.

The steps listed below can be used to compute a matrix’s inverse:

Step 1: Ascertain the minors of each matrix A member.

Step 2: Next, we determine the cofactors for each element, and then we construct the cofactor matrix by replacing each member of A with its corresponding cofactor.

Step 3: Transpose the cofactor matrix of A to determine its adjoint (written as adj A).

Step 4: Multiply adj A by the reciprocal of the determinant.

### Inverse of 3 x 3 Matrix

A few steps must be taken in order to find the Inverse Matrix Formula of a 3 by 3 matrix, but it is not an impossible task. A 3 x 3 matrix is made up of three rows and three columns.The numbers that make up the matrix are elements of the matrix. When the determinant is not equal to zero, a singular matrix exists. There is an Inverse Matrix Formula for each matrix of size m x m. M-1 stands in for it. Calculators cannot evaluate a matrix’s inverse, hence, it is improper to take shortcuts. An n x n matrix is possible if M is a non-singular square matrix. If M is a non-singular square matrix, then M-1, also known as the Inverse Matrix Formula of M, exists and has the following characteristics.

MM-1 = M-1 M = I

There is an Inverse Matrix Formula A-1 for each non-singular square matrix A such that A*A-1 = I. The three-square matrix given as A= a11, a12, a13

The Inverse Matrix Formula, A-1 = (1/|A|) Adj A, can be used to determine the Inverse Matrix Formula of a 3×3 matrix. Students must first determine whether the matrices are invertible, that is, whether |A| 0 exists.If a matrix has an inverse, they can calculate the adjoint of the supplied matrix and divide it by the matrix’s determinant. They may use a similar approach to determine the Inverse Matrix Formulaof any n n matrix.

To determine the Inverse Matrix Formula of a 333×3 matrix, follow these steps:

Determine the determinant of the provided matrix and whether it is invertible.

Determine the 2×2 minor matrices’ determinant.

Create the cofactor matrix.

To obtain the adjugate matrix, transpose the cofactor matrix.

Lastly, multiply the determinant by each term in the adjugate matrix.

### Inverse of 2 × 3 Matrix

Students must be aware that a square matrix must first exist in order for the Inverse Matrix Formula to exist. Additionally, this square matrix’s determinant must not be 0. The inverse of matrices of order m n will therefore not exist where m n. As a result, students are unable to determine the 2 by 3 matrix’s Inverse Matrix Formula. An Inverse Matrix Formula can only exist for non-singular matrices. A matrix’s determinant needs to be non-zero in order for it to be non-singular. Additionally, the determinant is restricted to square matrices. This suggests that the Inverse Matrix Formula of order m n will not exist where m x n. It is not possible to determine the Inverse Matrix Formula of the 2*3 matrix.

### Inverse of 2 × 1 Matrix

The supplied matrix is not a square matrix, just like the Inverse Matrix Formula of the 2*3 matrix;““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““ hence, the Inverse Matrix Formula of the 2*1 matrix similarly does not exist.

### Determinant of Inverse Matrix

The constituents of any row or column and their corresponding co-factors combine to form the matrix determinant. Only square matrices are provided for matrix determinants. Any square matrix A’s determinant is represented by the symbols det A (or) |A|. The symbol is sometimes used to denote it.

The inverse of the determinant of the original matrix is the determinant of the Inverse Matrix Formula.

Det(A • B) = Det (A) Det, as students are aware (B)

Det(A, B) = Det(A), Det(B), etc (B)

A A-1 Equals I as well.

A minus A-1 equals det (I)

Alternatively, det(A) = det(A-1) (I)

Because det(I) = 1,

A A-1 = det(A) det(A-1)

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

Thus, proved.

### Inverse of Matrix Examples

Analyze matrix A = (-3/2, 4/5) to find its inverse.

The matrices are as follows: A = (-3/2, 4/5)

The Inverse Matrix Formula  of the equation A = (a/c,b/d) can be calculated using the formula A-1 =1/ad-bc (d/-c, -b/a).

A-1 can be calculated using this formula as follows.

A-1= 1/ -3* 5* -4* 2 (5/-2, -4/-3)

=1/-15-8 (5/-2, -4/-3)

-1/23 (5/-2, -4/-3)

Consequently, A-1 is -1/23 (5/-2, -4/-33)

Practice Questions on Inverse of Matrix

1.Which of the following matrices has its inverse if ω, ω2 and are the cube roots of the units?

1. (1/w, w/w square)
2. (w square/1, 1/w)
3. (w/ w square, w square/ 1)
4. None of these

Answer: Option D. If a matrix’s determinant is not equal to 0, it has an inverse. The inverse does not exist for any of the given matrices since the determinants for options A, B, and C are all equal to zero.

2. How can a 33 3×3 matrix’s inverse be determined?

In order to determine the inverse of a 333×3  matrix, first determine the matrix’s determinant; if it is zero, the matrix does not exist. The matrix should then be rewritten with the first row becoming the first column, the middle row becoming the middle column, and the final row becoming the final column.