Question

Iffor a 10 x 10 matrix A, then the inequality$rank\left(A\right)⩽5$must hold.

Short answer

The above statement is true.

If for a 10 x 10 matrix A, then the inequality rank (A) ≤ 5 must hold.

Step-by-step solution

Mentioning the concept

If A and B are two matrices of order n x n then, we have

$rank\left(A\right)+rank\left(B\right)-n⩽rank\left(AB\right)$ (1)

Finding the rank of a matrix A of order 10 x 10

We have given a matrix A of order 10 x 10 such that_$A^2=0$

Let us take matrix A = matrix B

$\Rightarrow rank\left(A\right)=rank\left(B\right){\text{and rank(AB) = rank(A}}^2\text{)}$

$\because A^2=0\Rightarrow rank\left(A^2\right)=0$

Therefore, from inequality (1), we have

$$\begin{gathered} rank\left(A\right)+rank\left(A\right)-10⩽rank\left(A^2\right) \\ \Rightarrow 2rank\left(A\right)-10⩽0 \\ \Rightarrow rank\left(A\right)⩽5 \end{gathered}$$

Final Answer

If $A^2=0$for a 10 x 10 matrix A, then the inequality rank (A) ≤ 5 must hold.