Question

If A is an invertible n ×n matrix, then the kernels of A and $A^{-1}$must be equal.

Short answer

The above statement is true.

If A is an invertible n ×n matrix, then the kernels of A and $A^{-1}$must be equal.

Step-by-step solution

Definition of kernel of a matrix

Let A be a matrix of order m ×n, then the kernel of the matrix A, denoted by ker(A), is defined as-

$Ker\left(A\right)=\left\{x\in V:Ax=0\right\}$

To determine the kernel of a matrix A and A-1

Since A is an invertible n ×n matrix, then the rank of A and $A^{-1}$is ‘n’.

So, the nullity of both matrices A and $A^{-1}$is zero.

This implies that,

$Dim\left(\ker \left(A\right)\right)=Dim\left(\ker \left(A^{-1}\right)\right)=0$

$\Rightarrow \ker \left(A\right)=\overrightarrow{0},\ker \left(A^{-1}\right)=\overrightarrow{0}$

$\Rightarrow \ker \left(A\right)=\ker \left(A^{-1}\right)=\overrightarrow{0}$

Final Answer

If A is an invertible n ×n matrix, then

$Dim\left(\ker \left(A\right)\right)=Dim\left(\ker \left(A^{-1}\right)\right)=0$

$\Rightarrow \ker \left(A\right)=\ker \left(A^{-1}\right)=\overrightarrow{0}$

Hence, if A is an invertible n ×n matrix, then the kernels of A and $A^{-1}$must be equal.