Linear Algebra: 2.3 Invertible Matrices

An nxn matrix A is said to be invertible if there is an nxn matrix C such that CA = I and AC = I. What is the definition of the matrix C?

The inverse is _______ and denoted ______.

If ad - bc = 0, then A is invertible.

What is the determinant of A written as?

If A is an invertible nxn matrix, then for each b in R^n, what is the unique solution for x?

The product of nxn invertible matrices is invertible.

The inverse of AB is the product of the inverses of A and B in ________ order.

What is an elementary matrix?

A = nxn is invertible if and only if A is row equivalent to what?

What is the algorithm to find A^(-1) for a square matrix?

The matrix is invertible because its determinant is zero.

The matrix is not invertible because its columns are multiples of each other.

The matrix is invertible because it has three pivot positions.

If C is 6x6 and the equation Cx = v is consistent for every v in R^6, it is possible that for some v, the equation Cx = v has more than one solution.

If an nxn matrix K cannot be row reduced to I_n, what can you say about the columns of K?

Invertible Matrices and Their Properties

An nxn matrix A is invertible if there exists a matrix C such that CA = I and AC = I, where I is the identity matrix.
The inverse of matrix A is unique and is denoted by A^(-1).
If the determinant of a matrix A, calculated as ad - bc for a 2x2 matrix, equals zero (ad - bc = 0), then A is not invertible.
For any invertible matrix A and vector b in R^n, the equation Ax = b has a unique solution given by x = A^(-1)b.

Operations and Inverses

The product of two invertible matrices is also invertible.
The inverse of the product of two matrices AB is given by the product of their inverses in reverse order: (AB)^(-1) = B^(-1)A^(-1).

Elementary Matrices

An elementary matrix is derived by performing a single elementary row operation on an identity matrix.
A matrix A is invertible if and only if it is row equivalent to the identity matrix I_n. This means there exists a series of elementary row operations transforming A to I_n, which also implies I_n can be transformed to A^(-1).

Finding Inverses

To compute the inverse of a square matrix A, one can use the algorithm that reduces the augmented matrix [A | I] to [I | A^(-1)].

Determinants and Dependence

A matrix is invertible if its determinant is non-zero, indicating that its rows or columns are linearly independent.
If a matrix's columns are linearly dependent, meaning at least one column is a multiple of another, then the matrix is not invertible.
A matrix with three pivot positions in its row echelon form is guaranteed to be invertible.

Consistency and Solutions

For a 6x6 matrix C, if the equation Cx = v is consistent for every v in R^6, then it cannot have more than one solution for any v. This confirms that C is invertible, ensuring a unique solution exists for every consistent equation.
If an nxn matrix K cannot be reduced to the identity matrix I_n, it indicates that the columns of K are linearly dependent, thus they do not span R^n.

Test your understanding of invertible matrices through these flashcards. Learn the conditions for invertibility, the uniqueness of the matrix inverse, and the significance of the determinant. Perfect for students of Linear Algebra!