Inverse of a Matrix Formula

The inverse of matrix A, that is A^-1 is calculated using the inverse of matrix formula, which involves dividing the adjoint of a matrix by its determinant.

[Tex]A^{-1}=\frac{\text{Adj A}}{|A|}   [/Tex]

where,

• adj A = adjoint of the matrix A, and 
• |A| = determinant of the matrix A.

Note: This formula only works on Square matrices.

To find inverse of matrix using inverse of a matrix formula, follow these steps.

Step 1: Determine the minors of all A elements.

Step 2: Next, compute the cofactors of all elements and build the cofactor matrix by substituting the elements of A with their respective cofactors.

Step 3: Take the transpose of A’s cofactor matrix to find its adjoint (written as adj A).

Step 4: Multiply adj A by the reciprocal of the determinant of A.

Now, for any non-singular square matrix A,

A^-1 = 1 / |A| × Adj (A)

Example: Find the inverse of the matrix [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right][/Tex] using the formula.

We have, [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right] [/Tex]

Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose.

adj A = [Tex]\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex]

Find the value of determinant of the matrix.

|A| = 4(18–25) – 3(54–5) + 8(30–2)

⇒ |A| = 49

So, the inverse of the matrix is,

A^–1 = [Tex]\frac{1}{49}\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex]

⇒ A^–1 = [Tex]\left[\begin{array}{ccc}- \frac{1}{7} & \frac{13}{49} & – \frac{1}{49}\\-1 & \frac{4}{7} & \frac{4}{7}\\\frac{4}{7} & – \frac{17}{49} & – \frac{10}{49}\end{array}\right] [/Tex]

The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A^-1.

Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples.