in Mathematical Logic
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MX = O is a homogeneous equation and such an equation when |M| = 0 has non trivial solution.

M: Square Matrix

O: Null Matrix

Kindly help me with the above statement.
in Mathematical Logic
103 views

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$M$ is a square matrix, say of size $n \times n$.

Let $M$ has column vectors $m_1,m_2,...,m_n$ in $\mathbb{R}^n$.

$|M| = 0 \implies$ column vectors of $M$ are linearly dependent.

$\therefore c_1*m_1 + c_2*m_2 + ... + c_n*m_n = 0$, such that atleast one $c_i \neq 0$.

$\implies Mx = 0$ has non-trivial solution.

(Non-trivial means $x \neq 0$)
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When $|M| = 0$ then system $MX=0$ is having both trivial and non-trivial solutions. You will always get the trivial solution for such a system whether $|M|$ is zero or not.
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