Matrix Inversion Calculator

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Example

Created on 2024-06-20Asked by William White (Solvelet student)

Find the inverse of the matrix

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.

Solution

To find the inverse of the matrix

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},

we use the formula for the inverse of a 2x2 matrix:

A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

where

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

. 1. Compute the determinant of

A

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

2. Substitute

a, b, c,

and

d

into the formula:

A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}

3. Simplify the expression:

A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}

Therefore, the inverse of the matrix

A

is:

A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mateo Davis on Solvelet

1. Find the inverse of the matrix

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

.2. Determine whether the matrix

B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}

is invertible.

DefinitionMatrix Inversion: The matrix inversion is the process of finding the inverse of a square matrix A which is denoted as A−1 such that AA−1=A−1A=I Where I which is identity matrix. Only a few matrices are invertible; To be able to invert a matrix its determinant must be non-zero. A−1=−21(4−3−21)=(−2231−21).

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