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:
- Square (same number of rows and columns).
- 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=(acbd)
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
- Calculate the Determinant:
det(A)=ad−bctext{det}(A) = ad - bcdet(A)=ad−bc
- 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.
- 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=(1324).
- 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
- Check if the Matrix is Invertible:
Since det(A)=−2≠0text{det}(A) = -2 neq 0det(A)=−2=0, the matrix is invertible.
- 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)=(−2231−21)
Thus, the inverse of A=(1234)A = begin{pmatrix} 1 & 2 \ 3 & 4 end{pmatrix}A=(1324) is:
A−1=(−2132−12)A^{-1} = begin{pmatrix} -2 & 1 \ frac{3}{2} & -frac{1}{2} end{pmatrix}A−1=(−2231−21)
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=(acbd) 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.