Inverse matrix of order 2

Inverse matrix of order 2

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An inverse matrix is ​​the linear transformation of a matrix by multiplying the inverse of the determinant of the matrix by the adjoint transposed matrix.

In other words, an inverse matrix is ​​the multiplication of the inverse of the determinant by the transposed adjoint matrix.

Recommended articles: determinant of a matrix, square matrix, main diagonal and operations with matrices.

Given any matrix X such that

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Square matrix of order 2.

Inverse matrix formula of a matrix of order 2

Then the inverse matrix of X will be

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Inverse matrix formula of a square matrix of order 2.

Using this formula we obtain the inverse matrix of a square matrix of order 2.

The above formula can also be expressed by the determinant of the matrix.

Inverse matrix formula of a matrix of order 2

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Inverse matrix formula of a square matrix of order 2.

The two parallel lines around X in the denominator indicate that it is the determinant of the matrix X.

When a square matrix has an inverse matrix, we say that it is a regular matrix.

Requirements

To find the inverse matrix of a matrix of order n we need to meet the following requirements:

  • The matrix has to be a square matrix.

The number of rows (n) has to be the same as the number of columns (m). That is, the order of the matrix has to be n given that n = m.

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Square matrix of order n.
  • The determinant must be nonzero (0).

The determinant of the matrix must be nonzero (0) since it participates in the formula as a denominator. If the denominator were a zero (0) we would have an indeterminacy.

If the denominator (ad – bc) = 0, that is, the determinant of matrix X is equal to zero (0), then matrix X has no inverse matrix.

Property

A square matrix X of order n will have an inverse matrix X of order n, X -1 , such that it satisfies that

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Inverse matrix property.

The order of the elements of the multiplication is not relevant, that is, the multiplication of any square matrix by its inverse matrix will always result in the identity matrix of the same order.

In this case, the order of matrix X is 2. So, we can rewrite the previous property as:

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Inverse matrix property.

Practical example

Find the inverse matrix of matrix V.

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Example inverse matrix of order 2.

To solve this example we can apply the formula or first calculate the determinant and then substitute it.

Formula

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Application of the inverse matrix formula to matrix V.

Formula with determinant

We first calculate the determinant of the matrix V and then we substitute it into the formula.

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Determinant of matrix V.

Then, we obtain that the determinant of the matrix V is different from zero (0) and we can say that the matrix V does have an inverse matrix.

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Inverse matrix of matrix V by the determinant of V.

We obtain the same result using the formula or first calculating the determinant and then substituting it.

The order of the inverse matrix is ​​the same as the order of the original matrix. In this case, we will have the same number of rows n and columns m both in the matrix V and V -1 .

Matrix division

  • Square matrix
  • Attached matrix
  • Identity matrix