Question

Use matrix algebra to show that if A is invertible and D satisfies AD=1, then
D=A⁻¹

Answer 1

To show that if A is invertible and D satisfies AD=1, then D=A⁻¹ using matrix algebra, we can multiply both sides of the equation AD=1 by A⁻¹.

Step-by-step explanation:

To show that if A is invertible and D satisfies AD=1, then D=A⁻¹ using matrix algebra, we can multiply both sides of the equation AD=1 by A⁻¹. This gives:

A⁻¹(AD)=A⁻¹(1)

Simplifying the left side of the equation using the associative property of matrix multiplication, we have:

(A⁻¹A)D=A⁻¹

Since A⁻¹A is equal to the identity matrix I, the equation further simplifies to:

ID=A⁻¹

And since the identity matrix multiplied by any matrix equals the original matrix, we have the final result:

D=A⁻¹