An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. A matrix is invertible if and only if its determinant is non-zero.
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A square matrix $A$ is invertible if there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.
The inverse of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided that $ad - bc \neq 0$.
For an $n \times n$ matrix to be invertible, its determinant must be non-zero ($det(A) \neq 0$).
If a matrix can be row reduced to the identity matrix using elementary row operations, it is invertible.
The product of two invertible matrices is also an invertible matrix.
Review Questions
What condition must the determinant of a square matrix meet for it to be invertible?
How can you find the inverse of a 2x2 matrix?
What operation confirms that a given square matrix is indeed invertible?
A scalar value derived from a square matrix that indicates whether the matrix is invertible and encodes certain properties of linear transformations represented by the matrix.
A square matrix in which all elements of the principal diagonal are ones and all other elements are zeros. It acts as the multiplicative identity in matrix multiplication.
Operations including row swapping, scaling rows, and adding/subtracting multiples of rows used in solving systems of linear equations and finding inverses.