The n×n identity matrix I has ones down the main diagonal (upper-left corner to lower-right corner) and zeroes elsewhere. The 3×3 identity matrix, for example, is
The inverse of an n×n matrix A is an n×n matrix B satisfying AB = I. To find the inverse of an invertible matrix A, enter A with ``-1'' as a superscript and apply Evaluate. (As an alternative, leave the insertion point anywhere inside the matrix A, and from the Matrices submenu, choose Inverse.)
Evaluate
=
To check that this matrix satisfies the defining property, evaluate the product.
![]()
=
The operation Evaluate Numerically gives you a numerical approximation of the inverse. The accuracy of this numerical approximation depends on properties of the matrix, as well as on the choices you have made in the Settings menu.
Evaluate Numerically
=
Checking the product of a matrix with its inverse gives you an idea of the degree of accuracy of the approximation.
Evaluate
=
Since
An
=
A-1
, you can compute
negative powers of invertible matrices.
Evaluate
=
![]()
=
![]()
=
The preceding product demonstrates that
A-3 = A3
.
The m×n matrix with every entry equal to zero is the identity for addition; that is, for any m×n matrix A,
Evaluate
+
=
![]()