How To Find The Inverse Of A Matrix 2x2

Inverse of Matrix

Inverse of Matrix for a matrix A is denoted by A^-1. The changed of a 2 × 2 matrix can exist calculated using a simple formula. Farther, to discover the inverse of a 3 × 3 matrix, we need to know near 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 find the solution of linear equations through the matrix inversion method. Here, let us learn most the formula, methods, and terms related to the changed of matrix.

1.	What is Inverse of Matrix?
two.	Inverse of Matrix Formula
three.	Terms Related to Inverse of Matrix
4.	Methods to Observe Inverse of Matrix
5.	Determinant of Inverse Matrix
half dozen.	FAQs on Changed of Matrix

What is Inverse of Matrix?

The changed of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^-i, and A.A^-1 = A^-one·A = I, where I is the identity matrix. The matrix whose determinant is not-zip and for which the inverse matrix can exist calculated is called an invertible matrix. For instance, the changed of A = \(\left[\begin{assortment}{rr}
1 & -1 \\ \\
0 & 2
\terminate{array}\right]\) is \(\left[\brainstorm{array}{cc}
1 & 1 / 2 \\ \\
0 & i / two
\end{array}\right]\) as

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

Merely how to find the inverse of a matrix? Let usa run into in the upcoming sections.

Changed Matrix Formula

In the example of real numbers, the inverse of any real number a was the number a ^-1, such that a times a ^-one equals 1. Nosotros knew that for a real number, the changed of the number was the reciprocal of the number, as long equally the number wasn't zero. The inverse of a foursquare matrix A, denoted by A^-1, is the matrix and then that the production of A and A^-1 is the identity matrix. The identity matrix that results will be the same size as 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.

Changed 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 be a square matrix.
The determinant of the matrix should not be equal to naught.

The following terms beneath are helpful for more articulate understanding and easy adding of the inverse of matrix.

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

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

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

Cofactor of \(a_{ij}\) = (-one)^{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 cavalcade of the given matrix. The determinant of the matrix is equal to the summation of the production of the elements and its cofactors, of a particular row or column of the matrix.

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

Not-Singular Matrix: A matrix whose determinant value is not equal to zero is referred to equally a non-singular matrix. For a non-singular matrix |A| ≠ 0. A non-singular matrix is chosen an invertible matrix since its changed 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 cavalcade operations on determinants.

The value of the determinant remains unchanged if the rows and columns are interchanged.
The sign of the determinant changes, if whatever two rows or (two columns) are interchanged.
If whatever ii rows or columns of a matrix are equal, then the value of the determinant is zero.
If every element of a particular row or cavalcade is multiplied by a abiding, 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 exist expressed equally 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 cavalcade, then the value of the determinant remains unchanged.

Methods to Discover Changed of Matrix

The inverse of matrix can be found using two methods. The inverse of a matrix can be calculated through unproblematic 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 tin 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 nosotros use the matrix X and the second matrix B on the correct-mitt side of the equation.

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

Allow us cheque each of the methods described beneath.

Uncomplicated Row Operations

To summate the inverse of matrix A using unproblematic row transformations, nosotros first take the augmented matrix [A | I], where I is the identity matrix whose guild is the aforementioned as A. And so nosotros use the row operations to convert the left side A into I. And so the matrix gets converted into [I | A^-one]. For a more detailed process, click here.

Uncomplicated Column Operations

We tin can use the column operations also just similar how the process was explained for row operations to observe the changed 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 past the determinant of the matrix. The inverse of a matrix can be calculated by following the given steps:

Footstep i: Calculate the minors of all elements of A.
Step two: So compute cofactors of all elements and write the cofactor matrix by replacing the elements of A by their corresponding cofactors.
Pace 3: Notice the adjoint of A (written every bit adj A) past taking the tranpose of cofactor matrix of A.
Step four: Multiply adj A by reciprocal of determinant.

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

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

|A| = \(a_{eleven}(-one)^{1 + 1} \left|\brainstorm{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right| + a_{12}(-one)^{1 + ii} \left|\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right| + a_{13}(-ane)^{one + three} \left|\brainstorm{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\stop{matrix}\right|\)

Adj A = Transpose of Cofactor Matrix

= Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{xiii}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\stop{pmatrix}\)

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

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

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

Inverse of 2 × two 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 full general steps to calculate the inverse of a matrix. Let usa find the inverse of the 2 × 2 matrix given below:
A = \(\begin{bmatrix} a & b \\ \\ c & d \end{bmatrix}\)
A^-1 = (one/|A|) × Adj A
= [one/(advertisement - bc)] × \(\begin{bmatrix} d & -b \\ \\ -c & a \end{bmatrix}\)
Therefore, in order to calculate the inverse of 2 × 2 matrix, we need to first bandy the positions of terms a and d and put negative signs for terms b and c, and finally dissever 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 changed matrix A^-i, such that A × A^-1 = I. Permit united states of america take any 3 × 3 square matrix given as,

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

The inverse of 3x3 matrix can be calculated using the inverse matrix formula, A^-1 = (1/|A|) × Adj A

We volition first check if the given matrix is invertible, i.e., |A| ≠ 0. If the inverse of matrix exists, we can discover the adjoint of the given matrix and divide information technology past the determinant of the matrix.

The similar method can be followed to find the changed of whatsoever n × due north matrix. Let us encounter if similar steps tin be used to calculate the inverse of m × northward matrix.

Changed of ii × 3 Matrix

Nosotros know that the first condition for the inverse of a matrix to be is that the given matrix should be a square matrix. Likewise, the determinant of this square matrix should be non-cypher. This means that the inverse of matrices of the order m × north will not exist where one thousand ≠ north. Therefore, we cannot summate the inverse of 2 × three matrix.

Changed of 2 × ane Matrix

Similar to the changed of two × 3 matrix, the inverse of 2 × 1 matrix will likewise 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) = i / det(A). Let us bank check the proof of the higher up statement.

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

Also, A × A^-1 = I

det(A •A^-i) = det(I)

or, det(A) × det(A^-1) = det(I)

Since, det(I) = 1

det(A) × det(A^-1) = 1

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

Hence, proved.

Related Articles:

The post-obit related links are helpful in the better agreement of the changed of matrix.

Matrix Formula
Determinant Formula
Multiplication of Matrices

Important Points on Inverse of a Matrix:

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

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

Allow us run across how to apply the inverse matrix formula in the following solved examples section.

Inverse of Matrix Examples

Example 1: Notice the changed of matrix A = \(\left(\brainstorm{matrix}-3 & 4\\2 & 5 \finish{matrix}\right)\).

Solution:

The given matrix is A = \(\left(\begin{matrix}-three & 4\\ \\two & five \end{matrix}\correct)\).

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

Using this formula nosotros can summate A^-1 as follows.

A^-1 = \(\dfrac{1}{(-three)× 5 - iv × 2}\left(\begin{matrix}5&-iv\\\\-2&-3\end{matrix}\right)\)

= \(\dfrac{1}{-xv - 8}\left(\begin{matrix}5&-4\\\\-ii&-3\end{matrix}\right)\)

= \(\dfrac{-1}{23}\left(\begin{matrix}5&-4\\\\-2&-iii\end{matrix}\right)\)

Reply: Therefore A^-1 = \(\dfrac{-ane}{23}\left(\brainstorm{matrix}five&-4\\\\-2&-3\terminate{matrix}\right)\)
Case two: Find the changed of the matrix A = \(\left(\begin{matrix}4 & -ii & 1\\5&0&iii\\-1&2 & 6\terminate{matrix}\right)\).

Solution:

The given matrix is A = \(\left(\begin{matrix}iv & -2 & one\\5&0&3\\-1&2 & half dozen\end{matrix}\correct)\)

Pace - 1: Let us find the determinant of the given matrix using Row - ane of the above matrix.

|A| = \(4\left|\begin{matrix}0&three\\2 & 6\end{matrix}\correct| -(-2)\left|\begin{matrix}5&three\\-1 & 6\end{matrix}\correct|+i\left|\brainstorm{matrix}5&0\\-1& two\end{matrix}\right|\)

= 4(0 × 6 - 3 × 2) + ii(5 × half dozen - (-1) × 3) +1(v × 2 - 0 × (-i))

= 4(0 - half-dozen) + 2(thirty + 3) + ane(x - 0)

= -24 + 66 + 10

= 52

At present, 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.

Adj A = \(\left(\brainstorm{matrix}-vi & 14 & -6\\-33&25&-7\\ten&-vi & 10\end{matrix}\right)\)

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

A^-ane = \(\dfrac{ane}{52}\).\(\left(\begin{matrix}-half-dozen & fourteen & -6\\-33&25&-vii\\x&-half dozen & 10\end{matrix}\right)\)

= \(\left(\begin{matrix}-iii/26 & vii/26 & -3/26\\-33/52&25/52&-7/52\\5/26&-three/26 & 5/26\end{matrix}\right)\)

Reply: A^-1 = \(\left(\begin{matrix}-3/26 & 7/26 & -three/26\\-33/52&25/52&-vii/52\\5/26&-3/26 & 5/26\end{matrix}\right)\)
Example 3: Find the inverse of \(\begin{bmatrix}
4 & two \\\\
-1 & v
\terminate{bmatrix}\).

Solution:

To find: Changed of matrix \(\begin{bmatrix}
4 & ii \\\\
-one & 5
\finish{bmatrix}\)

Using the inverse of a matrix formula,

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

\( A^{-1} = \dfrac{1}{det \begin{pmatrix}4 & ii \\\\ -1 & five \end{pmatrix}} \begin{pmatrix}five & -two \\\\ ane & 4 \end{pmatrix}\)

Since, det \(\brainstorm{pmatrix}four & 2 \\\\ -1 & 5 \end{pmatrix}\) = 22

\( A^{-1} = \dfrac{1}{22} \begin{pmatrix}5 & -2 \\\\ ane & four \end{pmatrix} = \brainstorm{pmatrix} v/22 & -2/22 \\\\ one/22 & 4/22 \terminate{pmatrix} \)

Answer: Inverse of matrix \(\begin{bmatrix} 4 & ii \\ -1 & five \end{bmatrix} = \begin{bmatrix} 5/22 & -1/11 \\\\ 1/22 & ii/11 \end{bmatrix}\)

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Changed Matrix Questions

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FAQs on Inverse of Matrix

What is the Inverse of Matrix?

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

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

The inverse of matrix exists only if the determinant of the matrix is a not-zero value.

How to Find Changed of Matrix?

The inverse of a foursquare matrix is plant in two simple steps. First, the determinant and the adjoint of the given square matrix are calculated. Farther, the adjoint of the matrix is divided by the determinant to detect the inverse of the square matrix. The inverse of the matrix A is equal to \(\dfrac{1}{|A|}\).Adj A.

How to Observe Inverse of a 2 × 2 Matrix?

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

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

How to Use Inverse of Matrix?

The inverse of matrix is useful in solving equations by using the matrix inversion method. The matrix inversion method using the formula of X = A^-1B, where Ten is the variable matrix, A is the coefficient matrix, and B is the constant matrix.

Can Changed of Matrix be Calculated for an Invertible Matrix?

Yep, the inverse of matrix can be calculated for an invertible matrix. The matrix whose determinant is not equal to zero is a non-singular matrix. And for a nonsingular matrix, nosotros can find the determinant and the inverse of matrix.

When Does the Changed 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 foursquare matrix. Considering 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-zero value is called a non-singular matrix. Inverse is not divers for rectangular matrices.

What is the Formula for An Inverse Matrix?

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

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

For a given 2×ii matrix A = \(\left(\begin{matrix}a&b\\ \\c&d\end{matrix}\right)\) , changed is given by A^-i = \(\dfrac{one}{advertizing - bc}\left(\begin{matrix}d&-b\\\\-c&a\end{matrix}\right)\). Hither A^-1 is the changed of A.

How to Apply Inverse Matrix Formula?

The inverse matrix formula tin can exist used following the given steps:

Step 1: Find the matrix of minors for the given matrix.
Step 2: So notice the matrix of cofactors.
Pace iii: Discover the adjoint by taking the transpose of the matrix of cofactors.
Stride 4: Split it by the determinant.

What is iii × 3 Inverse Matrix Formula?

The inverse matrix formula for a iii × 3 matrix is, A^-ane = 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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