Invertible matrix

Invertible matrix refers to a square matrix that possesses an inverse. A matrix is square if it has the same number of rows and columns. An invertible matrix is also known as a nonsingular matrix or nondegenerate matrix. The inverse of a matrix A is denoted as A⁻¹, and it holds the property that when it is multiplied by A, it results in the identity matrix I. Mathematically, this is represented as AA⁻¹ = A⁻¹A = I, where I is the identity matrix of the same dimension as A.

Properties

Invertible matrices have several important properties:

• The determinant of an invertible matrix is non-zero. Conversely, if a matrix has a zero determinant, it is not invertible.
• The inverse of an invertible matrix is also invertible, and the inverse of the inverse is the original matrix.
• The product of two invertible matrices is itself invertible, with the inverse being the product of the inverses in the reverse order.
• The transpose of an invertible matrix is also invertible, and the inverse of the transpose is the transpose of the inverse.

Criteria for Invertibility

A square matrix is invertible if and only if it satisfies certain criteria:

• It must have a non-zero determinant.
• It must be of full rank, meaning all its rows and columns are linearly independent.
• There must exist a matrix B such that AB = BA = I, where I is the identity matrix.

Calculation Methods

Several methods exist for calculating the inverse of a matrix, if it exists:

• For 2x2 matrices, a direct formula can be applied.
• The Gaussian elimination method or its variants can be used for larger matrices.
• The adjugate matrix divided by the determinant also yields the inverse.

Applications

Invertible matrices are crucial in various fields such as linear algebra, computer graphics, and cryptography. They are used to solve systems of linear equations, perform geometric transformations, and encrypt or decrypt data, among other applications.

See Also

• Matrix
• Determinant
• Identity matrix
• Linear algebra

References

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