How To Find Inverse Of Any Matrix

Inverse of Matrix

Changed of Matrix for a matrix A is denoted by A^-ane. The inverse of a 2 × two matrix can exist calculated using a simple formula. Further, to discover the inverse of a 3 × iii matrix, we need to know nearly the determinant and adjoint of the matrix. The changed of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity.

The changed of matrix is used of notice 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.

1.	What is Changed of Matrix?
two.	Inverse of Matrix Formula
3.	Terms Related to Inverse of Matrix
iv.	Methods to Find Changed of Matrix
5.	Determinant of Inverse Matrix
half-dozen.	FAQs on Inverse of Matrix

What is Changed 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^-ane = A^-1·A = I, where I is the identity matrix. The matrix whose determinant is not-zero and for which the inverse matrix tin be calculated is called an invertible matrix. For example, the inverse of A = \(\left[\begin{assortment}{rr}
1 & -one \\ \\
0 & 2
\end{array}\right]\) is \(\left[\begin{assortment}{cc}
i & 1 / 2 \\ \\
0 & 1 / ii
\stop{array}\right]\) as

A.A^-1 = \(\left[\brainstorm{array}{rr}
1 & -1 \\ \\
0 & ii
\end{assortment}\right]\) \(\left[\brainstorm{assortment}{cc}
1 & 1 / 2 \\ \\
0 & i / ii
\end{assortment}\correct]\) = \(\left[\begin{array}{cc}
one & 0 \\ \\
0 & 1
\cease{assortment}\correct]\) = I
A^-1·A = \(\left[\brainstorm{array}{cc}
i & 1 / 2 \\ \\
0 & 1 / 2
\end{array}\right]\) \(\left[\begin{array}{rr}
ane & -1 \\ \\
0 & 2
\end{assortment}\right]\) = \(\left[\begin{array}{cc}
1 & 0 \\ \\
0 & i
\terminate{array}\right]\) = I

But how to find the changed of a matrix? Permit us see in the upcoming sections.

Inverse Matrix Formula

In the case of existent numbers, the inverse of whatever existent number a was the number a ^-ane, such that a times a ^-1 equals 1. Nosotros knew that for a existent number, the changed of the number was the reciprocal of the number, as long as the number wasn't nix. The inverse of a square matrix A, denoted by A^-ane, is the matrix so that the product of A and A^-1 is the identity matrix. The identity matrix that results will exist the aforementioned size equally matrix A.

Inverse of a Matrix

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 changed 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 be:

The given matrix should exist a square matrix.
The determinant of the matrix should non be equal to zero.

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

Small: The minor is divers for every element of a matrix. The minor of a particular element is the determinant obtained after eliminating the row and cavalcade containing this chemical element. For a matrix A = \(\begin{pmatrix} a_{11}&a_{12}&a_{thirteen}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the minor of the chemical element \(a_{xi}\) is:

Minor of \(a_{11}\) = \(\left|\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\cease{matrix}\right|\)

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

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

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 whatever 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 atypical matrix does not be.

Not-Atypical Matrix: A matrix whose determinant value is non equal to zero is referred to as a not-atypical 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 whatsoever ii rows or (two columns) are interchanged.
If whatsoever ii rows or columns of a matrix are equal, then the value of the determinant is cypher.
If every element of a item row or column is multiplied by a constant, and so the value of the determinant besides gets multiplied by the constant.
If the elements of a row or a column are expressed equally a sum of elements, then the determinant can exist expressed as a sum of determinants.
If the elements of a row or column are added or subtracted with the respective multiples of elements of another row or column, then the value of the determinant remains unchanged.

Methods to Find Inverse of Matrix

The changed of matrix can be plant using ii methods. The inverse of a matrix can be calculated through elementary operations and through the utilise of an adjoint of a matrix. The elementary operations on a matrix can be performed through row or column transformations. Too, the changed 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 unproblematic column operations nosotros apply the matrix Ten and the second matrix B on the right-manus side of the equation.

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

Allow us bank check each of the methods described beneath.

Elementary Row Operations

To summate the inverse of matrix A using elementary row transformations, we first accept the augmented matrix [A | I], where I is the identity matrix whose club is the aforementioned as A. And so we apply the row operations to convert the left side A into I. And then the matrix gets converted into [I | A^-1]. For a more than detailed process, click here.

Uncomplicated Column Operations

We can apply the column operations as well but like how the procedure 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, past dividing the adjoint of a matrix by the determinant of the matrix. The inverse of a matrix can exist calculated by following the given steps:

Step ane: Calculate the minors of all elements of A.
Step 2: And so compute cofactors of all elements and write the cofactor matrix by replacing the elements of A by their respective cofactors.
Step three: 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.

For a matrix A, its inverse A^-ane = \(\dfrac{1}{|A|}\)Adj A.

A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\terminate{pmatrix}\)

|A| = \(a_{11}(-ane)^{1 + i} \left|\brainstorm{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\finish{matrix}\right| + a_{12}(-1)^{i + 2} \left|\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\finish{matrix}\right| + a_{13}(-one)^{ane + 3} \left|\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\stop{matrix}\right|\)

Adj A = Transpose of Cofactor Matrix

= Transpose of \(\begin{pmatrix} A_{eleven}&A_{12}&A_{thirteen}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\)

=\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\finish{pmatrix}\)

A^-1 = \(\dfrac{1}{|A|}.\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{xiii}&A_{23}&A_{33}\terminate{pmatrix}\)

In this section, nosotros take learned the dissimilar methods to calculate the inverse of a matrix. Let us understand information technology better using a few examples for the different orders of matrices in the "examples" section below.

Inverse of 2 × two Matrix

The inverse of 2 × 2 matrix is easier to calculate in comparison to matrices of higher society. We tin calculate the changed of ii × ii matrix using the general steps to calculate the inverse of a matrix. Let us find the inverse of the ii × 2 matrix given beneath:
A = \(\brainstorm{bmatrix} a & b \\ \\ c & d \end{bmatrix}\)
A^-i = (1/|A|) × Adj A
= [1/(ad - bc)] × \(\begin{bmatrix} d & -b \\ \\ -c & a \end{bmatrix}\)
Therefore, in order to calculate the inverse of ii × 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 separate it by the determinant of the matrix.

Inverse of 3 × three Matrix

Nosotros know that for every not-singular square matrix A, in that location exists an inverse matrix A^-1, such that A × A^-i = I. Let usa take whatever 3 × 3 foursquare matrix given as,

A = \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\stop{bmatrix}\)

The inverse of 3x3 matrix tin be calculated using the changed matrix formula, A^-one = (one/|A|) × Adj A

Nosotros will first bank check if the given matrix is invertible, i.due east., |A| ≠ 0. If the changed 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 changed of whatsoever n × n matrix. Allow u.s. see if similar steps can be used to summate the inverse of one thousand × northward matrix.

Inverse of 2 × iii Matrix

Nosotros know that the first condition for the inverse of a matrix to be is that the given matrix should exist a square matrix. Also, the determinant of this square matrix should be non-goose egg. This ways that the inverse of matrices of the gild thou × due north will non exist where k ≠ n. Therefore, we cannot summate the changed of two × 3 matrix.

Inverse of 2 × 1 Matrix

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

Determinant of Changed Matrix

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

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

Also, A × A^-i = I

det(A •A^-one) = det(I)

or, det(A) × det(A^-1) = det(I)

Since, det(I) = i

det(A) × det(A^-1) = 1

or, det(A^-i) = i / det(A)

Hence, proved.

Topics Related to Inverse of Matrix:

The following related links are helpful in the ameliorate understanding of the inverse of matrix.

Matrix Formula
Determinant Formula
Multiplication of Matrices

Important Points on Changed of a Matrix:

The following points are helpful to empathise more clearly the thought of the inverse of matrix.

The inverse of a square matrix if exists, is unique.
If A and B are 2 invertible matrices of the aforementioned gild and then (AB)^-1 = B^-1A^-1.
The inverse of a foursquare matrix A exists, but if its determinant is a not-nix value, |A| ≠ 0.
The elements of a row or column, if multiplied with the cofactor elements of whatsoever other row or cavalcade, then their sum is nada.
The determinant of the product of ii matrices is equal to the product of the determinants of the two private matrices. |AB| = |A|.|B|

Let us see how to use the changed matrix formula in the post-obit solved examples section.

Solved Examples on Inverse of Matrix

Example 1: Find the inverse of matrix A = \(\left(\begin{matrix}-3 & four\\ii & v \end{matrix}\right)\).

Solution:

The given matrix is A = \(\left(\brainstorm{matrix}-iii & iv\\ \\2 & 5 \end{matrix}\right)\).

The formula to calculate the inverse of matrix for a matrix A = \(\left(\brainstorm{matrix}a&b\\\\c&d\stop{matrix}\correct)\) is A^-1 = \(\dfrac{1}{ad - bc}\left(\brainstorm{matrix}d&-b\\\\-c&a\finish{matrix}\right)\).

Using this formula we can summate A^-i as follows.

A^-1 = \(\dfrac{1}{(-3)× five - 4 × 2}\left(\brainstorm{matrix}5&-4\\\\-2&-3\end{matrix}\correct)\)

= \(\dfrac{1}{-fifteen - eight}\left(\begin{matrix}v&-4\\\\-2&-3\stop{matrix}\right)\)

= \(\dfrac{-1}{23}\left(\begin{matrix}five&-4\\\\-2&-3\stop{matrix}\right)\)

Reply: Therefore A^-1 = \(\dfrac{-i}{23}\left(\brainstorm{matrix}five&-iv\\\\-ii&-3\finish{matrix}\right)\)
Example two: Detect the inverse of the matrix A = \(\left(\brainstorm{matrix}4 & -ii & i\\5&0&iii\\-1&two & 6\end{matrix}\right)\).

Solution:

The given matrix is A = \(\left(\brainstorm{matrix}4 & -ii & i\\five&0&iii\\-1&2 & half-dozen\stop{matrix}\right)\)

Pace - i: Let the states observe the determinant of the given matrix using Row - 1 of the above matrix.

|A| = \(four\left|\brainstorm{matrix}0&3\\two & vi\end{matrix}\right| -(-2)\left|\brainstorm{matrix}five&3\\-one & half-dozen\finish{matrix}\right|+1\left|\begin{matrix}5&0\\-1& 2\end{matrix}\right|\)

= 4(0 × 6 - three × 2) + 2(5 × vi - (-ane) × 3) +ane(5 × 2 - 0 × (-i))

= 4(0 - six) + 2(30 + 3) + 1(x - 0)

= -24 + 66 + 10

= 52

At present, we volition decide the adjoint of the matrix A by computing the cofactors of each element and then taking the transpose of the cofactor matrix.

Adj A = \(\left(\brainstorm{matrix}-6 & fourteen & -6\\-33&25&-7\\x&-6 & 10\end{matrix}\correct)\)

The inverse of matrix A is given by the formula A^-1 = \(\dfrac{ane}{|A|}\).Adj A

A^-1 = \(\dfrac{ane}{52}\).\(\left(\brainstorm{matrix}-6 & 14 & -6\\-33&25&-7\\10&-half dozen & 10\end{matrix}\right)\)

= \(\left(\brainstorm{matrix}-3/26 & 7/26 & -3/26\\-33/52&25/52&-seven/52\\v/26&-iii/26 & five/26\end{matrix}\right)\)

Answer: A^-ane = \(\left(\brainstorm{matrix}-3/26 & 7/26 & -3/26\\-33/52&25/52&-7/52\\5/26&-iii/26 & 5/26\end{matrix}\right)\)
Example 3: Find the inverse of \(\begin{bmatrix}
4 & 2 \\\\
-ane & 5
\end{bmatrix}\).

Solution:

To detect: Inverse of matrix \(\begin{bmatrix}
4 & 2 \\\\
-1 & 5
\end{bmatrix}\)

Using the inverse of a matrix formula,

\( A^{-1} = \dfrac{\text{adj(A)}}{\text{|A|}}\)

\( A^{-i} = \dfrac{1}{det \begin{pmatrix}4 & two \\\\ -one & 5 \end{pmatrix}} \begin{pmatrix}5 & -2 \\\\ 1 & four \terminate{pmatrix}\)

Since, det \(\brainstorm{pmatrix}4 & 2 \\\\ -ane & 5 \end{pmatrix}\) = 22

\( A^{-1} = \dfrac{1}{22} \begin{pmatrix}five & -two \\\\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 5/22 & -2/22 \\\\ one/22 & 4/22 \cease{pmatrix} \)

Respond: Inverse of matrix \(\begin{bmatrix} 4 & 2 \\ -1 & 5 \end{bmatrix} = \begin{bmatrix} 5/22 & -1/11 \\\\ ane/22 & ii/11 \finish{bmatrix}\)

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Practice Questions on Inverse of Matrix

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FAQs on Inverse of Matrix

What is the Inverse of Matrix?

The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^-i, and A.A^-1 = I. The general formula for the inverse of matrix is equal to the adjoint of a matrix divided by the determinant of a matrix.

A^-ane = \(\dfrac{1}{|A|}\).Adj A

The changed of matrix exists just if the determinant of the matrix is a non-zip value.

How to Discover Inverse of Matrix?

The inverse of a square matrix is found in two uncomplicated steps. First, the determinant and the adjoint of the given foursquare matrix are calculated. Further, the adjoint of the matrix is divided past the determinant to find the inverse of the square matrix. The inverse of the matrix A is equal to \(\dfrac{1}{|A|}\).Adj A.

How to Find Inverse of a 2 × 2 Matrix?

The inverse of a 2 × 2 matrix is equal to the adjoint of the matrix divided by the determinant of the matrix. For a matrix A = \(\left(\begin{matrix}a&b\\ \\c&d\end{matrix}\correct)\), its adjoint is equal to the interchange of the elements of the offset diagonal and the sign change of the elements of the 2d diagonal. The formula for the inverse of the matrix is as follows.

A^-1 = \(\dfrac{i}{advertizement - bc}\left(\begin{matrix}d&-b\\\\-c&a\stop{matrix}\right)\)

How to Apply Inverse of Matrix?

The changed of matrix is useful in solving equations past using the matrix inversion method. The matrix inversion method using the formula of X = A^-iB, where Ten is the variable matrix, A is the coefficient matrix, and B is the constant matrix.

Tin can Changed of Matrix exist Calculated for an Invertible Matrix?

Yes, the changed of matrix tin be calculated for an invertible matrix. The matrix whose determinant is not equal to zero is a not-atypical matrix. And for a nonsingular matrix, we can discover the determinant and the inverse of matrix.

When Does the Inverse of Matrix Does not Exist in Some Cases?

The inverse of matrix exists only if its determinant value is a non-zero value and when the given matrix is a square matrix. Because the adjoint of the matrix is divided by the determinant of the matrix, to obtain the inverse of a matrix. The matrix whose determinant is a non-nada value is chosen a not-singular matrix. Changed is not defined for rectangular matrices.

What is the Formula for An Changed Matrix?

The changed matrix formula is used to determine the inverse matrix for any given matrix. The inverse of a square matrix, A is A^-1 only when: A × A^-ane = A^-ane × A = I. The inverse matrix formula can be given as, A^-ane = adj(A)/|A|; |A| ≠ 0, where A is a square matrix.

Given a two × 2 Matrix, What is the Formula for Finding the Inverse of the Matrix?

For a given two×2 matrix A = \(\left(\brainstorm{matrix}a&b\\ \\c&d\cease{matrix}\correct)\) , inverse is given past A^-one = \(\dfrac{ane}{ad - bc}\left(\brainstorm{matrix}d&-b\\\\-c&a\end{matrix}\correct)\). Here A^-1 is the inverse of A.

How to Utilize Inverse Matrix Formula?

The changed matrix formula tin can be used post-obit the given steps:

Step 1: Find the matrix of minors for the given matrix.
Step ii: Then observe the matrix of cofactors.
Step 3: Observe the adjoint by taking the transpose of the matrix of cofactors.
Step iv: Split information technology by the determinant.

What is 3 × iii Inverse Matrix Formula?

The inverse matrix formula for a three × iii matrix is, A^-1 = adj(A)/|A|; |A| ≠ 0 where A = square matrix, adj(A) = adjoint of square matrix, A^-one = inverse matrix

Source: https://www.cuemath.com/algebra/inverse-of-a-matrix/

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