Four matrices $${M_1},\,\,\,{M_2},\,\,\,{M_3}$$ and $${M_4}$$ of dimensions $$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,\,r\,\,x\,\,s$$ and $$\,\,\,\,s\,\,x\,\,t$$ respectively can be multiplied in sevaral ways with different number of total scalar multiplications. For example when multiplied as $$\left( {\left( {{M_1}\,\,X\,\,{M_2}} \right)\,\,X\,\,\left( {{M_3}\,\,X\,\,{M_4}} \right)} \right)$$, the total number of scalar multiplications is $$\,\,\,\,$$$$pqr + rst + prt$$. When multiplied as $$\left( {\left( {\left( {{M_1}\,\,X\,\,{M_2}} \right)\,\,X\,\,{M_3}} \right)X\,\,{M_4}} \right)$$, the total number of scalar multiplications is $$pqr + prs + pst$$. If $$p = 10,\,\,q = 100,\,\,r = 20,\,\,s = 5,\,\,$$ and $$t = 80$$, then the minimum number of scalar multiplications needed is

Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right]\,\,$$ If the eigen values of $$A$$ are $$4$$ and $$8$$, then

How many of the following matrices have an eigen value $$1$$?
$$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matrix{ 1 & { - 1} \cr 1 & 1 \cr } } \right]\,\,and\,\,\left[ {\matrix{ { - 1} & 0 \cr 1 & { - 1} \cr } } \right]$$

If $$M$$ is a square matrix with a zero determinant, which of the following assertion(s) is (are) correct?
$$S1$$ : Each row of $$M$$ can be represented as a linear combination of the other rows
$$S2$$ : Each column of $$M$$ can be represented as a linear combination of the other columns
$$S3$$ : $$MX$$ $$=$$ $$0$$ has a nontrivial solution
$$S4$$ : $$M$$ has an inverse