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Inverse of a Matrix

Definition

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. Not all matrices have an inverse; a matrix must be square (having the same number of rows and columns) and have a non-zero determinant to have an inverse.

If ( A ) is a square matrix, then its inverse, denoted as ( A^{-1} ), satisfies the following equation: [ A \cdot A^{-1} = A^{-1} \cdot A = I ] Where ( I ) is the identity matrix.

Conditions for Invertibility:

  • The matrix must be square (i.e., it has the same number of rows and columns).
  • The determinant of the matrix must be non-zero (( \text{det}(A) \neq 0 )).

The inverse of a 2x2 matrix can be calculated using a simple formula.

Formula for the Inverse of a 2x2 Matrix

Given a 2x2 matrix: [ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ] The inverse of ( A ), if it exists, is given by: [ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} ] Where ( \text{det}(A) = ad - bc ) is the determinant of the matrix.

The inverse exists only if ( \text{det}(A) \neq 0 ).

Example 1: Finding the Inverse of a 2x2 Matrix

Let ( A ) be the following matrix: [ A = \begin{bmatrix} 4 & 7 \ 2 & 6 \end{bmatrix} ]

Step 1: Calculate the Determinant

First, find the determinant of matrix ( A ): [ \text{det}(A) = (4 \cdot 6) - (7 \cdot 2) = 24 - 14 = 10 ]

Since ( \text{det}(A) = 10 \neq 0 ), the matrix is invertible.

Step 2: Apply the Inverse Formula

Now, apply the formula to find the inverse of ( A ): [ A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \ -2 & 4 \end{bmatrix} ]

Step 3: Simplify the Inverse

Multiply each element by ( \frac{1}{10} ): [ A^{-1} = \begin{bmatrix} 0.6 & -0.7 \ -0.2 & 0.4 \end{bmatrix} ]

Thus, the inverse of matrix ( A ) is: [ A^{-1} = \begin{bmatrix} 0.6 & -0.7 \ -0.2 & 0.4 \end{bmatrix} ]

Example 2: Finding the Inverse of Another 2x2 Matrix

Let ( B ) be the following matrix: [ B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ]

Step 1: Calculate the Determinant

First, find the determinant of matrix ( B ): [ \text{det}(B) = (1 \cdot 4) - (2 \cdot 3) = 4 - 6 = -2 ]

Since ( \text{det}(B) = -2 \neq 0 ), the matrix is invertible.

Step 2: Apply the Inverse Formula

Now, apply the formula to find the inverse of ( B ): [ B^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} ]

Step 3: Simplify the Inverse

Multiply each element by ( \frac{1}{-2} ): [ B^{-1} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} ]

Thus, the inverse of matrix ( B ) is: [ B^{-1} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} ]

Conclusion

  • The inverse of a matrix can be found only if the matrix is square and has a non-zero determinant.
  • The inverse of a 2x2 matrix is easily calculated using the formula involving the determinant and a rearrangement of the matrix elements.
  • If the determinant of a matrix is zero, the matrix does not have an inverse, and it is called singular.

Finding the inverse of a matrix is crucial in solving systems of linear equations, transforming matrices, and performing other operations in linear algebra.

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