Question-83

system of equations

matrix

determinant

Choose the correct options from the following:

Let $\displaystyle A,B$ be square matrices of the same order such that $\displaystyle AB=0$. Then at least one of the matrices must have determinant $\displaystyle 0$.
If $\displaystyle A-\alpha I=0$ for a square matrix $\displaystyle A$ of order $\displaystyle n$ and $\displaystyle \alpha \in \mathbb{R}$, the determinant of $\displaystyle A$ is always positive.
A matrix with all diagonal entries zero will have a zero determinant.
For a square matrix $\displaystyle A$, if the system of linear equations $\displaystyle Ax=0$ has a unique solution, then the determinant of $\displaystyle A$ is non-zero.

Answer

Let $\displaystyle A,B$ be square matrices of the same order such that $\displaystyle AB=0$. Then at least one of the matrices must have determinant $\displaystyle 0$.
If $\displaystyle A-\alpha I=0$ for a square matrix $\displaystyle A$ of order $\displaystyle n$ and $\displaystyle \alpha \in \mathbb{R}$, the determinant of $\displaystyle A$ is always positive.
A matrix with all diagonal entries zero will have a zero determinant.
For a square matrix $\displaystyle A$, if the system of linear equations $\displaystyle Ax=0$ has a unique solution, then the determinant of $\displaystyle A$ is non-zero.

Solution

Option-1

If $\displaystyle AB=0$, then $\displaystyle |AB|=0$. This implies that $\displaystyle |A|\cdot |B|=0$, so the determinant of at least one of these two matrices has to be zero.

Option-2

We have:

\[ \begin{equation*} \begin{aligned} |A| & =|\alpha I|\\ & =\alpha ^{n} \end{aligned} \end{equation*} \]

The determinant of $\displaystyle A$ will be positive for any arbitrary $$ only if $\displaystyle n$ is even.

Option-3

Here is a counter example. $\displaystyle A=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$. $\displaystyle |A|=-1$.

Option-4

If $\displaystyle Ax=0$ has a unique solution for some square matrix $\displaystyle A$, the columns of $\displaystyle A$ are linearly independent and hence $\displaystyle |A|\neq 0$.