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Matrix Inverse calculator finds the inverse of a given matrix: A-1, with detailed step by step calculations.
Matrix Inverse Calculator finds the inverse of given matrix.
Steps to find the inverse of given matrix A
By the definition of inverse of a matrix, we can write
A.A-1 = A-1.A = I
And, by the definition of an identity matrix I, we can write
I.A = A.I = A
Consider the following equation.
A = I.A
Make row transformations to the matrix A on left hand side, and the identity matrix I on the right side, such that it transforms into the following form.
I = A-1.A
With row transformations, if we can transform the left side matrix A into Identity matrix, then the identity matrix on right hand side should have been transformed to Inverse of matrix A, which is what we needed.
Examples to use Matrix Inverse Calculator
1. Find the inverse of the following 3×3 matrix.
Matrix \({\ A = \begin{bmatrix}1 & 3 & 3\\1 & 4 & 3\\1 & 3 & 4\end{bmatrix} }\)
Solution
Since the given matrix is of size 3×3, follow these steps with the Matrix Inverse Calculator.
- Enter matrix size of 3×3 in input fields: m=3 and n=3.
- A matrix with specified size of 3×3 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 3 & 3\\1 & 4 & 3\\1 & 3 & 4\end{bmatrix} \)
Computing Matrix Inverse: \( A^{-1} \)
We know that
\( A = I \cdot A\)
Applying \( R1 → (1)R1\)
\( \begin{bmatrix}1 & 3 & 3\\1 & 4 & 3\\1 & 3 & 4\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R2 → R2-(1)R1 \)
\( \begin{bmatrix}1 & 3 & 3\\0 & 1 & 0\\1 & 3 & 4\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\-1 & 1 & 0\\0 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R3 → R3-(1)R1 \)
\( \begin{bmatrix}1 & 3 & 3\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R2 → (1)R2\)
\( \begin{bmatrix}1 & 3 & 3\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R1 → R1-(3)R2 \)
\( \begin{bmatrix}1 & 0 & 3\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}4 & -3 & 0\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R3 → (1)R3\)
\( \begin{bmatrix}1 & 0 & 3\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}4 & -3 & 0\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \cdot A\)
Applying \( R1 → R1-(3)R3 \)
\( \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}7 & -3 & -3\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \cdot A\)
The above equation is in the form \(I = A^{-1}.A\), where \(A^{-1} = \begin{bmatrix}7 & -3 & -3\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \).
Result
Therefore, the inverse of the given matrix A,
\(A^{-1} = \begin{bmatrix}7 & -3 & -3\\-1 & 1 & 0\\-1 & 0 & 1\end{bmatrix} \)
2. Find the inverse of the following 2×2 matrix.
Matrix \({\ A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} }\)
Solution
Since the given matrix is of size 2×2, follow these steps with the Matrix Inverse Calculator.
- Enter matrix size of 2×2 in input fields: m=2 and n=2.
- A matrix with specified size of 2×2 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} \)
Computing Matrix Inverse: \( A^{-1} \)
We know that
\( A = I \cdot A\)
Applying \( R1 → (1)R1\)
\( \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} \cdot A\)
Applying \( R2 → R2-(3)R1 \)
\( \begin{bmatrix}1 & 2\\0 & -2\end{bmatrix} = \begin{bmatrix}1 & 0\\-3 & 1\end{bmatrix} \cdot A\)
Applying \( R2 → (\frac{-1}{2})R2\)
\( \begin{bmatrix}1 & 2\\0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0\\\frac{3}{2} & \frac{-1}{2}\end{bmatrix} \cdot A\)
Applying \( R1 → R1-(2)R2 \)
\( \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} = \begin{bmatrix}-2 & 1\\\frac{3}{2} & \frac{-1}{2}\end{bmatrix} \cdot A\)
The above equation is in the form \(I = A^{-1}.A\), where \(A^{-1} = \begin{bmatrix}-2 & 1\\\frac{3}{2} & \frac{-1}{2}\end{bmatrix} \).
Result
Therefore, the inverse of the given matrix A,
\(A^{-1} = \begin{bmatrix}-2 & 1\\\frac{3}{2} & \frac{-1}{2}\end{bmatrix} \)