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Matrix InverseCalculator

Matrix A size: x





Answer




How to use this Matrix Inverse Calculator 🤔

Follow these steps to perform Matrix Inverse for the given matrices.

  1. Enter the matrix size for matrix A.
  2. Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
  3. As and when you complete entering the matrix elements, Matrix Inverse is calculated, and displayed in the answer section.

Matrix Inverse

The inverse of a matrix A is another matrix, denoted as A-1, such that when A is multiplied by A-1, the result is the identity matrix. The matrix A must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.

The inverse of a matrix A is computed as:

\( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \)

where \( \det(A) \) is the determinant of A and \( \text{adj}(A) \) is the adjugate of A.

Example 1

Let's find the inverse of the following 2x2 matrix:

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)

The determinant of A is:

\( det(A) = (1 \times 4) - (2 \times 3) = 4 - 6 = -2 \)

The adjugate of A is:

\( \text{adj}(A) = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \)

Therefore, the inverse of A is:

\( A^{-1} = \frac{1}{-2} \times \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \)