Determine for all values of the parameter $p$ whether the statement $d∈<a,b,c>$ holds for the vectors $a, b, c, d ∈ R^4:$

Question

Determine for all values of the parameter $p$ whether the statement $d∈<a,b,c>$ holds for the vectors $a, b, c, d ∈ R^4:$ $a=3,−1,2,1)^T$ , $b=(15,8,8,7)^T$ , $c=(12,6,7,p)^T$ , $d=(6,8,−9,12)^T$

$d$ is a subspace of $a,b,c$ ? How can we find all values of the parameter $p?$

Answer 1

Hint: first verify that a $4\times 3$ matrix whose columns are $a,b,c$ is non-singular. Then define another matrix ,$A$, whose columns are $a,b,c,d$ and find values of $p$ for which $|A|=0$ (what is the intuition behind that?)