Question

The product of any matrix by the scalar is the null matrix.

Answer 1

Multiply scalar ' 0 ' with an $n\times n$ matrix A Scalar multiplication of a matrix is obtained by multiplying each element of the matrix by that scalar.
Let A be an $n\times n$ matrix

Then, $0\cdot A=0$$0.\left[\begin{array}{ccc}{a}_{11}& a_{12}\dots \dots & a_{1n}\\ ⋮& & \\ a_{n1}& a_{n2}\dots \dots & a_{nn}\end{array}\right]$

$=\left[\begin{array}{cccc}0& 0\dots \dots & 0& \\ ⋮& & & \\ 0& 0\dots \dots & 0& \end{array}\right]$

$=0$

Hence, the product of any matrix by the scalar 0 is the null matrix.