Invertible Matrix

Definition

An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse. This means that when the matrix is multiplied by its inverse, the result is the identity matrix. The existence of an inverse indicates that the matrix represents a one-to-one linear transformation, which is crucial in solving systems of linear equations and understanding linear transformations.

5 Must Know Facts For Your Next Test

1. A square matrix is invertible if and only if its determinant is non-zero.
2. The inverse of a matrix A is denoted as A^{-1}, and it satisfies the equation AA^{-1} = A^{-1}A = I, where I is the identity matrix.
3. Not all square matrices are invertible; those with linearly dependent rows or columns will be singular and have no inverse.
4. If a matrix is invertible, its row operations can be performed without losing information about the solution set of a linear system.
5. The process to find an inverse often involves row reduction or using the adjugate and determinant, depending on the method chosen.

Review Questions

• How can you determine if a square matrix is invertible using its determinant?
  • To determine if a square matrix is invertible, you compute its determinant. If the determinant is non-zero, then the matrix is invertible, meaning it has an inverse. Conversely, if the determinant equals zero, the matrix is singular and does not have an inverse. This relationship is key in understanding the properties of linear transformations represented by the matrix.
• Describe how an invertible matrix relates to solving systems of linear equations.
  • An invertible matrix plays a crucial role in solving systems of linear equations. When representing a system as Ax = b, where A is the coefficient matrix, if A is invertible, we can find a unique solution for x by calculating x = A^{-1}b. This capability highlights how an invertible matrix ensures that every input vector in the solution space corresponds to a unique output vector, preserving the one-to-one relationship needed for reliable solutions.
• Evaluate the significance of row operations in relation to finding the inverse of an invertible matrix and what implications this has for linear transformations.
  • Row operations are essential for finding the inverse of an invertible matrix, as they allow us to systematically reduce the augmented matrix [A | I] to obtain A^{-1}. The ability to perform these operations without losing information about the system indicates that every linear transformation represented by an invertible matrix maintains a one-to-one correspondence between inputs and outputs. This property is significant because it guarantees that any linear system associated with the invertible matrix can be uniquely solved and understood in terms of its geometric transformation.

Related terms

Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere, acting as the multiplicative identity in matrix multiplication.

Determinant:

A scalar value that can be computed from a square matrix, which provides important information about the matrix, including whether it is invertible.

Singular Matrix:

A square matrix that does not have an inverse, typically because its determinant is zero.