Inverse of a Matrix

The inverse of a matrix AAA is another matrix A−1A^{-1}A−1 such that:

A⋅A−1=A−1⋅A=IA cdot A^{-1} = A^{-1} cdot A = IA⋅A−1=A−1⋅A=I

where III is the identity matrix. The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere.

A matrix must be:

1. Square (same number of rows and columns).

2. Non-Singular (having a non-zero determinant) to have an inverse.

Inverse of a 2x2 Matrix

For a 2x2 matrix AAA:

A=(abcd)A = begin{pmatrix} a & b \ c & d end{pmatrix}A=(ac​bd​)

the inverse A−1A^{-1}A−1 is calculated using the formula:

A−1=1det(A)(d−b−ca)A^{-1} = frac{1}{text{det}(A)} begin{pmatrix} d & -b \ -c & a end{pmatrix}A−1=det(A)1​(d−c​−ba​)

where det(A)=ad−bctext{det}(A) = ad - bcdet(A)=ad−bc is the determinant of the matrix AAA.

Steps to Find the Inverse

1. Calculate the Determinant:

det(A)=ad−bctext{det}(A) = ad - bcdet(A)=ad−bc

1. Check if the Matrix is Invertible:

Ensure det(A)≠0text{det}(A) neq 0det(A)=0. If det(A)=0text{det}(A) = 0det(A)=0, the matrix is singular and does not have an inverse.

1. Apply the Formula:

If det(A)≠0text{det}(A) neq 0det(A)=0, use the formula to find the inverse.

Example

Consider the matrix A=(1234)A = begin{pmatrix} 1 & 2 \ 3 & 4 end{pmatrix}A=(13​24​).

1. Calculate the Determinant:

det(A)=1⋅4−2⋅3=4−6=−2text{det}(A) = 1 cdot 4 - 2 cdot 3 = 4 - 6 = -2det(A)=1⋅4−2⋅3=4−6=−2

1. Check if the Matrix is Invertible:

Since det(A)=−2≠0text{det}(A) = -2 neq 0det(A)=−2=0, the matrix is invertible.

1. Apply the Formula:

A−1=1−2(4−2−31)=(−2132−12)A^{-1} = frac{1}{-2} begin{pmatrix} 4 & -2 \ -3 & 1 end{pmatrix} = begin{pmatrix} -2 & 1 \ frac{3}{2} & -frac{1}{2} end{pmatrix}A−1=−21​(4−3​−21​)=(−223​​1−21​​)

Thus, the inverse of A=(1234)A = begin{pmatrix} 1 & 2 \ 3 & 4 end{pmatrix}A=(13​24​) is:

A−1=(−2132−12)A^{-1} = begin{pmatrix} -2 & 1 \ frac{3}{2} & -frac{1}{2} end{pmatrix}A−1=(−223​​1−21​​)

General Method for Larger Matrices

For larger matrices (greater than 2x2), the inverse can be found using methods such as:

• Gauss-Jordan Elimination: Augment the matrix with the identity matrix and perform row operations to transform the original matrix into the identity matrix. The augmented part becomes the inverse.

• Adjugate and Determinant: Calculate the cofactor matrix, take its transpose (adjugate), and divide by the determinant.

• LU Decomposition: Decompose the matrix into a product of a lower triangular matrix and an upper triangular matrix, and use these to find the inverse.

Summary

• The inverse of a matrix AAA is denoted A−1A^{-1}A−1 and satisfies A⋅A−1=IA cdot A^{-1} = IA⋅A−1=I.

• A matrix must be square and non-singular to have an inverse.

• The inverse of a 2x2 matrix A=(abcd)A = begin{pmatrix} a & b \ c & d end{pmatrix}A=(ac​bd​) is found using A−1=1det(A)(d−b−ca)A^{-1} = frac{1}{text{det}(A)} begin{pmatrix} d & -b \ -c & a end{pmatrix}A−1=det(A)1​(d−c​−ba​).

• For larger matrices, the inverse can be found using Gauss-Jordan elimination, adjugate and determinant, or LU decomposition methods.