Summarize this article:
Last updated on September 26, 2025
The inverse of a matrix A can be represented as A–1. We use a direct formula to calculate the inverse of a 2 × 2 matrix. For matrices of order 3 or higher, we calculate the determinant and the adjoint to find the inverse. The matrix inversion method is useful for solving linear equations effectively. In this article, we will discuss the inverse of a matrix and its wider applications in real life.
The inverse of a matrix is what results in the identity matrix when multiplied by the original. Note that a matrix can have an inverse only if it has a non-zero determinant, and such matrices are called invertible matrices.
For example:
The inverse of a square matrix A, denoted as A⁻¹, is a matrix that, when multiplied with A, yields the identity matrix. If A is a square matrix, then its inverse satisfies A × A⁻¹ = A⁻¹ × A = I
So, the inverse is:
We represent the inverse of any square matrix A as A–1. Additionally, the product of a matrix and its inverse always results in the identity matrix I.
The inverse of a square matrix A is given by the formula:
A–1= 1A× adj (A)
Where:
|A| → determinant of matrix A
Adj (A) → adjoint or transpose of cofactors of A
What are the properties of Inverse of a Matrix?
The inverse of the matrix has certain key features, as discussed below:
The inverse of an inverse matrix is the original matrix.
(A–1)-1= A
The inverse of the product of two matrices, A and B, is represented as.
(AB)-1= B–1 A–1
A matrix only possesses an inverse if it is non-singular.
The inverse of the transpose of a matrix is equal to the transpose of its inverse:
(AT)-1 = (A-1)T
How to Find the Matrix Inverse?
The two common methods to determine the inverse of a matrix are:
Using Matrix Formula:
The inverse of a matrix is the adjoint divided by its determinant.
A–1 (inverse of a matrix A) = Adj A/ |A|
To find the matrix inverse, we use two Inverse matrix methods, such as:
Determinant Method:
The inverse of a matrix is calculated using the determinant and adjoint.
The formula for a square matrix A:
A–1 = 1/ det (A) × adj (A)
Where:
A–1 → inverse of matrix A
det (A)→ determinant of A
adj(A) → adjoint of A
If the determinant of A is zero, the matrix has no inverse.
Elementary Transformation Method:
In this method, we find the inverse using a step-by-step process:
Step 1: Express the matrix equation as [A\I]. Here, I is the identity matrix, which is in the same order as the matrix A.
Step 2: To convert matrix A into the identity matrix on the left, use only row operations or only column operations. Apply the same step to the identity matrix on the right side.
The matrix B on the right is the inverse of A once you obtain: I = B × A.
Step 3: Use either row operations or column operations exclusively, not both simultaneously.
Using this method, we can quickly determine the inverse of a 2 × 2 matrix.
2 × 2 matrix Inverse
The inverse of a 2 × 2 matrix can be found using a simple method:
Let’s say:
Since A-1 = 1/ (ad – bc)
Using the formula:
A–1= (1/ |A|) × adj A
So,
Hence, the inverse of the 2 × 2 matrix is determined.
Inverse of a 3 × 3 Matrix
In the case of any 3 × 3 matrix:
Use the inverse matrix formula:
A–1= 1/ | A| × adj (A)
Determinant of Inverse Matrix
The determinant of an inverse matrix is derived by taking the reciprocal of the original matrix’s determinant: det(A⁻¹) = 1/det(A)
Proof (in simple steps):
We know:
det(A × B) = det(A) × det(B)
Also,
A × A⁻¹ = I (by inverse matrix property)
So,
det(A × A⁻¹) = det(I)
det(A) × det(A⁻¹) = 1 (since det(I) = 1)
det(A⁻¹) = 1/det(A)
Thus, the determinant of the inverse is the reciprocal of the original determinant, as shown.
The inverse of a matrix is not just a mathematical concept; it has numerous practical uses beyond math. Here are a few real-life instances where it can be applied.
The inverse of a matrix is an important concept in mathematics. However, students often make mistakes while calculating it. Here are a few common mistakes and tips to avoid them.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.